[{"data":1,"prerenderedAt":15207},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-06-06-proof-of-the-uat-by-weierstrass-theory":3},{"id":4,"title":5,"body":6,"cardimage":15193,"description":15183,"draft":15194,"enableComment":15195,"extension":15196,"image":15183,"meta":15197,"navigation":15195,"onday":15198,"path":15199,"seo":15200,"stem":15201,"summary":15202,"tags":15203,"__hash__":15206},"blog\u002Fblog\u002F2026\u002F2026-06-06-proof-of-the-uat-by-weierstrass-theory.md","Proof of the UAT, by Weierstrass Theory",{"type":7,"value":8,"toc":15182},"minimark",[9,14,19,23,154,239,242,300,351,549,552,558,561,1123,1126,1129,1346,1378,1381,1484,1487,1493,1496,1936,1939,2041,2044,2245,2248,2566,2569,2572,2575,2734,2737,3058,3061,3369,3372,3545,3592,3595,3793,4100,4102,4239,4270,4326,4489,4492,4705,4707,5046,5049,5053,5220,5310,5313,5499,5502,5899,5902,6089,6092,6246,6249,6438,6578,6813,6919,7073,7250,7421,7424,7867,7869,8171,8203,8430,8432,8609,8640,8643,8914,8916,9026,9029,9032,9036,9039,9201,9539,9542,9728,9761,9766,9813,10013,10281,10347,10452,10455,10778,10826,11020,11023,11391,11394,11576,11833,11895,11904,11909,12286,12556,12559,12811,12814,13467,13472,13475,13725,13728,14070,14073,14076,14080,14083,14086,14648,14651,14954,14957,15179],[10,11,13],"h2",{"id":12},"weierstrass-theorem","Weierstrass Theorem",[15,16,18],"h3",{"id":17},"intuitive-explanation","Intuitive Explanation",[20,21,22],"p",{},"Suppose",[24,25,29],"span",{"className":26,"translate":28},[27],"katex-display","no",[24,30,33,88],{"className":31,"translate":28},[32],"katex",[24,34,37],{"className":35},[36],"katex-mathml",[38,39,42],"math",{"xmlns":40,"display":41},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[43,44,45,83],"semantics",{},[46,47,48,52,56,59,63,66,70,74,77,80],"mrow",{},[49,50,51],"mi",{},"f",[53,54,55],"mo",{},"∈",[49,57,58],{},"C",[53,60,62],{"stretchy":61},"false","(",[53,64,65],{"stretchy":61},"[",[67,68,69],"mn",{},"0",[53,71,73],{"separator":72},"true",",",[67,75,76],{},"1",[53,78,79],{"stretchy":61},"]",[53,81,82],{"stretchy":61},")",[84,85,87],"annotation",{"encoding":86},"application\u002Fx-tex","f\\in C([0,1])",[24,89,92,119],{"className":90,"ariaHidden":72},[91],"katex-html",[24,93,96,101,107,112,116],{"className":94},[95],"base",[24,97],{"className":98,"style":100},[99],"strut","height:0.8889em;vertical-align:-0.1944em;",[24,102,51],{"className":103,"style":106},[104,105],"mord","mathnormal","margin-right:0.1076em;",[24,108],{"className":109,"style":111},[110],"mspace","margin-right:0.2778em;",[24,113,55],{"className":114},[115],"mrel",[24,117],{"className":118,"style":111},[110],[24,120,122,126,130,135,138,142,146,149],{"className":121},[95],[24,123],{"className":124,"style":125},[99],"height:1em;vertical-align:-0.25em;",[24,127,58],{"className":128,"style":129},[104,105],"margin-right:0.0715em;",[24,131,134],{"className":132},[133],"mopen","([",[24,136,69],{"className":137},[104],[24,139,73],{"className":140},[141],"mpunct",[24,143],{"className":144,"style":145},[110],"margin-right:0.1667em;",[24,147,76],{"className":148},[104],[24,150,153],{"className":151},[152],"mclose","])",[20,155,156,157,185,186,238],{},"i.e., ",[24,158,160,173],{"className":159,"translate":28},[32],[24,161,163],{"className":162},[36],[38,164,165],{"xmlns":40},[43,166,167,171],{},[46,168,169],{},[49,170,51],{},[84,172,51],{"encoding":86},[24,174,176],{"className":175,"ariaHidden":72},[91],[24,177,179,182],{"className":178},[95],[24,180],{"className":181,"style":100},[99],[24,183,51],{"className":184,"style":106},[104,105]," is continuous on ",[24,187,189,211],{"className":188,"translate":28},[32],[24,190,192],{"className":191},[36],[38,193,194],{"xmlns":40},[43,195,196,208],{},[46,197,198,200,202,204,206],{},[53,199,65],{"stretchy":61},[67,201,69],{},[53,203,73],{"separator":72},[67,205,76],{},[53,207,79],{"stretchy":61},[84,209,210],{"encoding":86},"[0,1]",[24,212,214],{"className":213,"ariaHidden":72},[91],[24,215,217,220,223,226,229,232,235],{"className":216},[95],[24,218],{"className":219,"style":125},[99],[24,221,65],{"className":222},[133],[24,224,69],{"className":225},[104],[24,227,73],{"className":228},[141],[24,230],{"className":231,"style":145},[110],[24,233,76],{"className":234},[104],[24,236,79],{"className":237},[152],".",[20,240,241],{},"Then for any",[24,243,245],{"className":244,"translate":28},[27],[24,246,248,268],{"className":247,"translate":28},[32],[24,249,251],{"className":250},[36],[38,252,253],{"xmlns":40,"display":41},[43,254,255,265],{},[46,256,257,260,263],{},[49,258,259],{},"ε",[53,261,262],{},">",[67,264,69],{},[84,266,267],{"encoding":86},"\\varepsilon>0",[24,269,271,290],{"className":270,"ariaHidden":72},[91],[24,272,274,278,281,284,287],{"className":273},[95],[24,275],{"className":276,"style":277},[99],"height:0.5782em;vertical-align:-0.0391em;",[24,279,259],{"className":280},[104,105],[24,282],{"className":283,"style":111},[110],[24,285,262],{"className":286},[115],[24,288],{"className":289,"style":111},[110],[24,291,293,297],{"className":292},[95],[24,294],{"className":295,"style":296},[99],"height:0.6444em;",[24,298,69],{"className":299},[104],[20,301,302,303,350],{},"there exists a polynomial ",[24,304,306,328],{"className":305,"translate":28},[32],[24,307,309],{"className":308},[36],[38,310,311],{"xmlns":40},[43,312,313,325],{},[46,314,315,318,320,323],{},[49,316,317],{},"P",[53,319,62],{"stretchy":61},[49,321,322],{},"x",[53,324,82],{"stretchy":61},[84,326,327],{"encoding":86},"P(x)",[24,329,331],{"className":330,"ariaHidden":72},[91],[24,332,334,337,341,344,347],{"className":333},[95],[24,335],{"className":336,"style":125},[99],[24,338,317],{"className":339,"style":340},[104,105],"margin-right:0.1389em;",[24,342,62],{"className":343},[133],[24,345,322],{"className":346},[104,105],[24,348,82],{"className":349},[152]," such that",[24,352,354],{"className":353,"translate":28},[27],[24,355,357,423],{"className":356,"translate":28},[32],[24,358,360],{"className":359},[36],[38,361,362],{"xmlns":40,"display":41},[43,363,364,420],{},[46,365,366,370,372,374,376,378,381,383,385,387,389,391,394,396,398,401,404,406,408,410,412,414,416,418],{},[49,367,369],{"mathvariant":368},"normal","∣",[49,371,51],{},[53,373,62],{"stretchy":61},[49,375,322],{},[53,377,82],{"stretchy":61},[53,379,380],{},"−",[49,382,317],{},[53,384,62],{"stretchy":61},[49,386,322],{},[53,388,82],{"stretchy":61},[49,390,369],{"mathvariant":368},[53,392,393],{},"\u003C",[49,395,259],{},[53,397,73],{"separator":72},[110,399],{"width":400},"2em",[49,402,403],{"mathvariant":368},"∀",[49,405,322],{},[53,407,55],{},[53,409,65],{"stretchy":61},[67,411,69],{},[53,413,73],{"separator":72},[67,415,76],{},[53,417,79],{"stretchy":61},[49,419,238],{"mathvariant":368},[84,421,422],{"encoding":86},"|f(x)-P(x)|\u003C\\varepsilon,\n\\qquad\n\\forall x\\in [0,1].",[24,424,426,458,488,522],{"className":425,"ariaHidden":72},[91],[24,427,429,432,435,438,441,444,447,451,455],{"className":428},[95],[24,430],{"className":431,"style":125},[99],[24,433,369],{"className":434},[104],[24,436,51],{"className":437,"style":106},[104,105],[24,439,62],{"className":440},[133],[24,442,322],{"className":443},[104,105],[24,445,82],{"className":446},[152],[24,448],{"className":449,"style":450},[110],"margin-right:0.2222em;",[24,452,380],{"className":453},[454],"mbin",[24,456],{"className":457,"style":450},[110],[24,459,461,464,467,470,473,476,479,482,485],{"className":460},[95],[24,462],{"className":463,"style":125},[99],[24,465,317],{"className":466,"style":340},[104,105],[24,468,62],{"className":469},[133],[24,471,322],{"className":472},[104,105],[24,474,82],{"className":475},[152],[24,477,369],{"className":478},[104],[24,480],{"className":481,"style":111},[110],[24,483,393],{"className":484},[115],[24,486],{"className":487,"style":111},[110],[24,489,491,494,497,500,504,507,510,513,516,519],{"className":490},[95],[24,492],{"className":493,"style":100},[99],[24,495,259],{"className":496},[104,105],[24,498,73],{"className":499},[141],[24,501],{"className":502,"style":503},[110],"margin-right:2em;",[24,505],{"className":506,"style":145},[110],[24,508,403],{"className":509},[104],[24,511,322],{"className":512},[104,105],[24,514],{"className":515,"style":111},[110],[24,517,55],{"className":518},[115],[24,520],{"className":521,"style":111},[110],[24,523,525,528,531,534,537,540,543,546],{"className":524},[95],[24,526],{"className":527,"style":125},[99],[24,529,65],{"className":530},[133],[24,532,69],{"className":533},[104],[24,535,73],{"className":536},[141],[24,538],{"className":539,"style":145},[110],[24,541,76],{"className":542},[104],[24,544,79],{"className":545},[152],[24,547,238],{"className":548},[104],[20,550,551],{},"Next, let's try to prove this theorem.",[20,553,554],{},[555,556,557],"strong",{},"Step 1: Construct the Bernstein polynomial",[20,559,560],{},"Define",[24,562,564],{"className":563,"translate":28},[27],[24,565,567,681],{"className":566,"translate":28},[32],[24,568,570],{"className":569},[36],[38,571,572],{"xmlns":40,"display":41},[43,573,574,678],{},[46,575,576,585,587,589,591,593,595,597,600,617,619,623,636,649,656,658,660,662,664,676],{},[577,578,579,582],"msub",{},[49,580,581],{},"B",[49,583,584],{},"n",[53,586,62],{"stretchy":61},[49,588,51],{},[53,590,82],{"stretchy":61},[53,592,62],{"stretchy":61},[49,594,322],{},[53,596,82],{"stretchy":61},[53,598,599],{},"=",[601,602,603,606,615],"munderover",{},[53,604,605],{},"∑",[46,607,608,611,613],{},[49,609,610],{},"k",[53,612,599],{},[67,614,69],{},[49,616,584],{},[49,618,51],{},[620,621,622],"mtext",{}," ⁣",[46,624,625,627,634],{},[53,626,62],{"fence":72},[628,629,630,632],"mfrac",{},[49,631,610],{},[49,633,584],{},[53,635,82],{"fence":72},[46,637,638,640,647],{},[53,639,62],{"fence":72},[628,641,643,645],{"linethickness":642},"0px",[49,644,584],{},[49,646,610],{},[53,648,82],{"fence":72},[650,651,652,654],"msup",{},[49,653,322],{},[49,655,610],{},[53,657,62],{"stretchy":61},[67,659,76],{},[53,661,380],{},[49,663,322],{},[650,665,666,668],{},[53,667,82],{"stretchy":61},[46,669,670,672,674],{},[49,671,584],{},[53,673,380],{},[49,675,610],{},[49,677,238],{"mathvariant":368},[84,679,680],{"encoding":86},"B_n(f)(x)\n=\n\n\\sum_{k=0}^{n}\nf\\!\\left(\\frac{k}{n}\\right)\n\\binom{n}{k}\nx^k(1-x)^{n-k}.",[24,682,684,774,1072],{"className":683,"ariaHidden":72},[91],[24,685,687,690,747,750,753,756,759,762,765,768,771],{"className":686},[95],[24,688],{"className":689,"style":125},[99],[24,691,693,697],{"className":692},[104],[24,694,581],{"className":695,"style":696},[104,105],"margin-right:0.0502em;",[24,698,701],{"className":699},[700],"msupsub",[24,702,706,738],{"className":703},[704,705],"vlist-t","vlist-t2",[24,707,710,733],{"className":708},[709],"vlist-r",[24,711,715],{"className":712,"style":714},[713],"vlist","height:0.1514em;",[24,716,718,723],{"style":717},"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;",[24,719],{"className":720,"style":722},[721],"pstrut","height:2.7em;",[24,724,730],{"className":725},[726,727,728,729],"sizing","reset-size6","size3","mtight",[24,731,584],{"className":732},[104,105,729],[24,734,737],{"className":735},[736],"vlist-s","​",[24,739,741],{"className":740},[709],[24,742,745],{"className":743,"style":744},[713],"height:0.15em;",[24,746],{},[24,748,62],{"className":749},[133],[24,751,51],{"className":752,"style":106},[104,105],[24,754,82],{"className":755},[152],[24,757,62],{"className":758},[133],[24,760,322],{"className":761},[104,105],[24,763,82],{"className":764},[152],[24,766],{"className":767,"style":111},[110],[24,769,599],{"className":770},[115],[24,772],{"className":773,"style":111},[110],[24,775,777,781,859,862,865,869,872,962,965,1026,1057,1060,1063,1066,1069],{"className":776},[95],[24,778],{"className":779,"style":780},[99],"height:2.9535em;vertical-align:-1.3021em;",[24,782,786],{"className":783},[784,785],"mop","op-limits",[24,787,789,850],{"className":788},[704,705],[24,790,792,847],{"className":791},[709],[24,793,796,819,832],{"className":794,"style":795},[713],"height:1.6514em;",[24,797,799,803],{"style":798},"top:-1.8479em;margin-left:0em;",[24,800],{"className":801,"style":802},[721],"height:3.05em;",[24,804,806],{"className":805},[726,727,728,729],[24,807,809,813,816],{"className":808},[104,729],[24,810,610],{"className":811,"style":812},[104,105,729],"margin-right:0.0315em;",[24,814,599],{"className":815},[115,729],[24,817,69],{"className":818},[104,729],[24,820,822,825],{"style":821},"top:-3.05em;",[24,823],{"className":824,"style":802},[721],[24,826,827],{},[24,828,605],{"className":829},[784,830,831],"op-symbol","large-op",[24,833,835,838],{"style":834},"top:-4.3em;margin-left:0em;",[24,836],{"className":837,"style":802},[721],[24,839,841],{"className":840},[726,727,728,729],[24,842,844],{"className":843},[104,729],[24,845,584],{"className":846},[104,105,729],[24,848,737],{"className":849},[736],[24,851,853],{"className":852},[709],[24,854,857],{"className":855,"style":856},[713],"height:1.3021em;",[24,858],{},[24,860],{"className":861,"style":145},[110],[24,863,51],{"className":864,"style":106},[104,105],[24,866],{"className":867,"style":868},[110],"margin-right:-0.1667em;",[24,870],{"className":871,"style":145},[110],[24,873,876,885,956],{"className":874},[875],"minner",[24,877,881],{"className":878,"style":880},[133,879],"delimcenter","top:0em;",[24,882,62],{"className":883},[884,728],"delimsizing",[24,886,888,892,953],{"className":887},[104],[24,889],{"className":890},[133,891],"nulldelimiter",[24,893,895],{"className":894},[628],[24,896,898,944],{"className":897},[704,705],[24,899,901,941],{"className":900},[709],[24,902,905,918,929],{"className":903,"style":904},[713],"height:1.3714em;",[24,906,908,912],{"style":907},"top:-2.314em;",[24,909],{"className":910,"style":911},[721],"height:3em;",[24,913,915],{"className":914},[104],[24,916,584],{"className":917},[104,105],[24,919,921,924],{"style":920},"top:-3.23em;",[24,922],{"className":923,"style":911},[721],[24,925],{"className":926,"style":928},[927],"frac-line","border-bottom-width:0.04em;",[24,930,932,935],{"style":931},"top:-3.677em;",[24,933],{"className":934,"style":911},[721],[24,936,938],{"className":937},[104],[24,939,610],{"className":940,"style":812},[104,105],[24,942,737],{"className":943},[736],[24,945,947],{"className":946},[709],[24,948,951],{"className":949,"style":950},[713],"height:0.686em;",[24,952],{},[24,954],{"className":955},[152,891],[24,957,959],{"className":958,"style":880},[152,879],[24,960,82],{"className":961},[884,728],[24,963],{"className":964,"style":145},[110],[24,966,968,974,1020],{"className":967},[104],[24,969,971],{"className":970,"style":880},[133,879],[24,972,62],{"className":973},[884,728],[24,975,977],{"className":976},[628],[24,978,980,1012],{"className":979},[704,705],[24,981,983,1009],{"className":982},[709],[24,984,987,998],{"className":985,"style":986},[713],"height:1.1076em;",[24,988,989,992],{"style":907},[24,990],{"className":991,"style":911},[721],[24,993,995],{"className":994},[104],[24,996,610],{"className":997,"style":812},[104,105],[24,999,1000,1003],{"style":931},[24,1001],{"className":1002,"style":911},[721],[24,1004,1006],{"className":1005},[104],[24,1007,584],{"className":1008},[104,105],[24,1010,737],{"className":1011},[736],[24,1013,1015],{"className":1014},[709],[24,1016,1018],{"className":1017,"style":950},[713],[24,1019],{},[24,1021,1023],{"className":1022,"style":880},[152,879],[24,1024,82],{"className":1025},[884,728],[24,1027,1029,1032],{"className":1028},[104],[24,1030,322],{"className":1031},[104,105],[24,1033,1035],{"className":1034},[700],[24,1036,1038],{"className":1037},[704],[24,1039,1041],{"className":1040},[709],[24,1042,1045],{"className":1043,"style":1044},[713],"height:0.8991em;",[24,1046,1048,1051],{"style":1047},"top:-3.113em;margin-right:0.05em;",[24,1049],{"className":1050,"style":722},[721],[24,1052,1054],{"className":1053},[726,727,728,729],[24,1055,610],{"className":1056,"style":812},[104,105,729],[24,1058,62],{"className":1059},[133],[24,1061,76],{"className":1062},[104],[24,1064],{"className":1065,"style":450},[110],[24,1067,380],{"className":1068},[454],[24,1070],{"className":1071,"style":450},[110],[24,1073,1075,1079,1082,1120],{"className":1074},[95],[24,1076],{"className":1077,"style":1078},[99],"height:1.1491em;vertical-align:-0.25em;",[24,1080,322],{"className":1081},[104,105],[24,1083,1085,1088],{"className":1084},[152],[24,1086,82],{"className":1087},[152],[24,1089,1091],{"className":1090},[700],[24,1092,1094],{"className":1093},[704],[24,1095,1097],{"className":1096},[709],[24,1098,1100],{"className":1099,"style":1044},[713],[24,1101,1102,1105],{"style":1047},[24,1103],{"className":1104,"style":722},[721],[24,1106,1108],{"className":1107},[726,727,728,729],[24,1109,1111,1114,1117],{"className":1110},[104,729],[24,1112,584],{"className":1113},[104,105,729],[24,1115,380],{"className":1116},[454,729],[24,1118,610],{"className":1119,"style":812},[104,105,729],[24,1121,238],{"className":1122},[104],[20,1124,1125],{},"This is the Bernstein polynomial.",[20,1127,1128],{},"Note that",[24,1130,1132],{"className":1131,"translate":28},[27],[24,1133,1135,1185],{"className":1134,"translate":28},[32],[24,1136,1138],{"className":1137},[36],[38,1139,1140],{"xmlns":40,"display":41},[43,1141,1142,1182],{},[46,1143,1144,1156,1162,1164,1166,1168,1170],{},[46,1145,1146,1148,1154],{},[53,1147,62],{"fence":72},[628,1149,1150,1152],{"linethickness":642},[49,1151,584],{},[49,1153,610],{},[53,1155,82],{"fence":72},[650,1157,1158,1160],{},[49,1159,322],{},[49,1161,610],{},[53,1163,62],{"stretchy":61},[67,1165,76],{},[53,1167,380],{},[49,1169,322],{},[650,1171,1172,1174],{},[53,1173,82],{"stretchy":61},[46,1175,1176,1178,1180],{},[49,1177,584],{},[53,1179,380],{},[49,1181,610],{},[84,1183,1184],{"encoding":86},"\\binom{n}{k}\nx^k(1-x)^{n-k}",[24,1186,1188,1299],{"className":1187,"ariaHidden":72},[91],[24,1189,1191,1195,1255,1284,1287,1290,1293,1296],{"className":1190},[95],[24,1192],{"className":1193,"style":1194},[99],"height:2.4em;vertical-align:-0.95em;",[24,1196,1198,1204,1249],{"className":1197},[104],[24,1199,1201],{"className":1200,"style":880},[133,879],[24,1202,62],{"className":1203},[884,728],[24,1205,1207],{"className":1206},[628],[24,1208,1210,1241],{"className":1209},[704,705],[24,1211,1213,1238],{"className":1212},[709],[24,1214,1216,1227],{"className":1215,"style":986},[713],[24,1217,1218,1221],{"style":907},[24,1219],{"className":1220,"style":911},[721],[24,1222,1224],{"className":1223},[104],[24,1225,610],{"className":1226,"style":812},[104,105],[24,1228,1229,1232],{"style":931},[24,1230],{"className":1231,"style":911},[721],[24,1233,1235],{"className":1234},[104],[24,1236,584],{"className":1237},[104,105],[24,1239,737],{"className":1240},[736],[24,1242,1244],{"className":1243},[709],[24,1245,1247],{"className":1246,"style":950},[713],[24,1248],{},[24,1250,1252],{"className":1251,"style":880},[152,879],[24,1253,82],{"className":1254},[884,728],[24,1256,1258,1261],{"className":1257},[104],[24,1259,322],{"className":1260},[104,105],[24,1262,1264],{"className":1263},[700],[24,1265,1267],{"className":1266},[704],[24,1268,1270],{"className":1269},[709],[24,1271,1273],{"className":1272,"style":1044},[713],[24,1274,1275,1278],{"style":1047},[24,1276],{"className":1277,"style":722},[721],[24,1279,1281],{"className":1280},[726,727,728,729],[24,1282,610],{"className":1283,"style":812},[104,105,729],[24,1285,62],{"className":1286},[133],[24,1288,76],{"className":1289},[104],[24,1291],{"className":1292,"style":450},[110],[24,1294,380],{"className":1295},[454],[24,1297],{"className":1298,"style":450},[110],[24,1300,1302,1305,1308],{"className":1301},[95],[24,1303],{"className":1304,"style":1078},[99],[24,1306,322],{"className":1307},[104,105],[24,1309,1311,1314],{"className":1310},[152],[24,1312,82],{"className":1313},[152],[24,1315,1317],{"className":1316},[700],[24,1318,1320],{"className":1319},[704],[24,1321,1323],{"className":1322},[709],[24,1324,1326],{"className":1325,"style":1044},[713],[24,1327,1328,1331],{"style":1047},[24,1329],{"className":1330,"style":722},[721],[24,1332,1334],{"className":1333},[726,727,728,729],[24,1335,1337,1340,1343],{"className":1336},[104,729],[24,1338,584],{"className":1339},[104,105,729],[24,1341,380],{"className":1342},[454,729],[24,1344,610],{"className":1345,"style":812},[104,105,729],[20,1347,1348,1349,238],{},"is itself a polynomial in ",[24,1350,1352,1365],{"className":1351,"translate":28},[32],[24,1353,1355],{"className":1354},[36],[38,1356,1357],{"xmlns":40},[43,1358,1359,1363],{},[46,1360,1361],{},[49,1362,322],{},[84,1364,322],{"encoding":86},[24,1366,1368],{"className":1367,"ariaHidden":72},[91],[24,1369,1371,1375],{"className":1370},[95],[24,1372],{"className":1373,"style":1374},[99],"height:0.4306em;",[24,1376,322],{"className":1377},[104,105],[20,1379,1380],{},"Therefore",[24,1382,1384],{"className":1383,"translate":28},[27],[24,1385,1387,1417],{"className":1386,"translate":28},[32],[24,1388,1390],{"className":1389},[36],[38,1391,1392],{"xmlns":40,"display":41},[43,1393,1394,1414],{},[46,1395,1396,1402,1404,1406,1408,1410,1412],{},[577,1397,1398,1400],{},[49,1399,581],{},[49,1401,584],{},[53,1403,62],{"stretchy":61},[49,1405,51],{},[53,1407,82],{"stretchy":61},[53,1409,62],{"stretchy":61},[49,1411,322],{},[53,1413,82],{"stretchy":61},[84,1415,1416],{"encoding":86},"B_n(f)(x)",[24,1418,1420],{"className":1419,"ariaHidden":72},[91],[24,1421,1423,1426,1466,1469,1472,1475,1478,1481],{"className":1422},[95],[24,1424],{"className":1425,"style":125},[99],[24,1427,1429,1432],{"className":1428},[104],[24,1430,581],{"className":1431,"style":696},[104,105],[24,1433,1435],{"className":1434},[700],[24,1436,1438,1458],{"className":1437},[704,705],[24,1439,1441,1455],{"className":1440},[709],[24,1442,1444],{"className":1443,"style":714},[713],[24,1445,1446,1449],{"style":717},[24,1447],{"className":1448,"style":722},[721],[24,1450,1452],{"className":1451},[726,727,728,729],[24,1453,584],{"className":1454},[104,105,729],[24,1456,737],{"className":1457},[736],[24,1459,1461],{"className":1460},[709],[24,1462,1464],{"className":1463,"style":744},[713],[24,1465],{},[24,1467,62],{"className":1468},[133],[24,1470,51],{"className":1471,"style":106},[104,105],[24,1473,82],{"className":1474},[152],[24,1476,62],{"className":1477},[133],[24,1479,322],{"className":1480},[104,105],[24,1482,82],{"className":1483},[152],[20,1485,1486],{},"is also a polynomial.",[20,1488,1489,1492],{},[555,1490,1491],{},"Step 2: Understand what it does."," This polynomial is a bit abstract, but we can interpret it in another way.",[20,1494,1495],{},"First note the identity",[24,1497,1499],{"className":1498,"translate":28},[27],[24,1500,1502,1595],{"className":1501,"translate":28},[32],[24,1503,1505],{"className":1504},[36],[38,1506,1507],{"xmlns":40,"display":41},[43,1508,1509,1592],{},[46,1510,1511,1525,1537,1543,1545,1547,1549,1551,1563,1565,1567,1569,1572,1574,1576,1578,1580,1582,1588,1590],{},[601,1512,1513,1515,1523],{},[53,1514,605],{},[46,1516,1517,1519,1521],{},[49,1518,610],{},[53,1520,599],{},[67,1522,69],{},[49,1524,584],{},[46,1526,1527,1529,1535],{},[53,1528,62],{"fence":72},[628,1530,1531,1533],{"linethickness":642},[49,1532,584],{},[49,1534,610],{},[53,1536,82],{"fence":72},[650,1538,1539,1541],{},[49,1540,322],{},[49,1542,610],{},[53,1544,62],{"stretchy":61},[67,1546,76],{},[53,1548,380],{},[49,1550,322],{},[650,1552,1553,1555],{},[53,1554,82],{"stretchy":61},[46,1556,1557,1559,1561],{},[49,1558,584],{},[53,1560,380],{},[49,1562,610],{},[53,1564,599],{},[53,1566,62],{"stretchy":61},[49,1568,322],{},[53,1570,1571],{},"+",[53,1573,62],{"stretchy":61},[67,1575,76],{},[53,1577,380],{},[49,1579,322],{},[53,1581,82],{"stretchy":61},[650,1583,1584,1586],{},[53,1585,82],{"stretchy":61},[49,1587,584],{},[53,1589,599],{},[67,1591,76],{},[84,1593,1594],{"encoding":86},"\\sum_{k=0}^{n}\n\\binom{n}{k}\nx^k(1-x)^{n-k}\n=(x+(1-x))^n\n=1",[24,1596,1598,1778,1834,1855,1876,1927],{"className":1597,"ariaHidden":72},[91],[24,1599,1601,1604,1671,1674,1734,1763,1766,1769,1772,1775],{"className":1600},[95],[24,1602],{"className":1603,"style":780},[99],[24,1605,1607],{"className":1606},[784,785],[24,1608,1610,1663],{"className":1609},[704,705],[24,1611,1613,1660],{"className":1612},[709],[24,1614,1616,1636,1646],{"className":1615,"style":795},[713],[24,1617,1618,1621],{"style":798},[24,1619],{"className":1620,"style":802},[721],[24,1622,1624],{"className":1623},[726,727,728,729],[24,1625,1627,1630,1633],{"className":1626},[104,729],[24,1628,610],{"className":1629,"style":812},[104,105,729],[24,1631,599],{"className":1632},[115,729],[24,1634,69],{"className":1635},[104,729],[24,1637,1638,1641],{"style":821},[24,1639],{"className":1640,"style":802},[721],[24,1642,1643],{},[24,1644,605],{"className":1645},[784,830,831],[24,1647,1648,1651],{"style":834},[24,1649],{"className":1650,"style":802},[721],[24,1652,1654],{"className":1653},[726,727,728,729],[24,1655,1657],{"className":1656},[104,729],[24,1658,584],{"className":1659},[104,105,729],[24,1661,737],{"className":1662},[736],[24,1664,1666],{"className":1665},[709],[24,1667,1669],{"className":1668,"style":856},[713],[24,1670],{},[24,1672],{"className":1673,"style":145},[110],[24,1675,1677,1683,1728],{"className":1676},[104],[24,1678,1680],{"className":1679,"style":880},[133,879],[24,1681,62],{"className":1682},[884,728],[24,1684,1686],{"className":1685},[628],[24,1687,1689,1720],{"className":1688},[704,705],[24,1690,1692,1717],{"className":1691},[709],[24,1693,1695,1706],{"className":1694,"style":986},[713],[24,1696,1697,1700],{"style":907},[24,1698],{"className":1699,"style":911},[721],[24,1701,1703],{"className":1702},[104],[24,1704,610],{"className":1705,"style":812},[104,105],[24,1707,1708,1711],{"style":931},[24,1709],{"className":1710,"style":911},[721],[24,1712,1714],{"className":1713},[104],[24,1715,584],{"className":1716},[104,105],[24,1718,737],{"className":1719},[736],[24,1721,1723],{"className":1722},[709],[24,1724,1726],{"className":1725,"style":950},[713],[24,1727],{},[24,1729,1731],{"className":1730,"style":880},[152,879],[24,1732,82],{"className":1733},[884,728],[24,1735,1737,1740],{"className":1736},[104],[24,1738,322],{"className":1739},[104,105],[24,1741,1743],{"className":1742},[700],[24,1744,1746],{"className":1745},[704],[24,1747,1749],{"className":1748},[709],[24,1750,1752],{"className":1751,"style":1044},[713],[24,1753,1754,1757],{"style":1047},[24,1755],{"className":1756,"style":722},[721],[24,1758,1760],{"className":1759},[726,727,728,729],[24,1761,610],{"className":1762,"style":812},[104,105,729],[24,1764,62],{"className":1765},[133],[24,1767,76],{"className":1768},[104],[24,1770],{"className":1771,"style":450},[110],[24,1773,380],{"className":1774},[454],[24,1776],{"className":1777,"style":450},[110],[24,1779,1781,1784,1787,1825,1828,1831],{"className":1780},[95],[24,1782],{"className":1783,"style":1078},[99],[24,1785,322],{"className":1786},[104,105],[24,1788,1790,1793],{"className":1789},[152],[24,1791,82],{"className":1792},[152],[24,1794,1796],{"className":1795},[700],[24,1797,1799],{"className":1798},[704],[24,1800,1802],{"className":1801},[709],[24,1803,1805],{"className":1804,"style":1044},[713],[24,1806,1807,1810],{"style":1047},[24,1808],{"className":1809,"style":722},[721],[24,1811,1813],{"className":1812},[726,727,728,729],[24,1814,1816,1819,1822],{"className":1815},[104,729],[24,1817,584],{"className":1818},[104,105,729],[24,1820,380],{"className":1821},[454,729],[24,1823,610],{"className":1824,"style":812},[104,105,729],[24,1826],{"className":1827,"style":111},[110],[24,1829,599],{"className":1830},[115],[24,1832],{"className":1833,"style":111},[110],[24,1835,1837,1840,1843,1846,1849,1852],{"className":1836},[95],[24,1838],{"className":1839,"style":125},[99],[24,1841,62],{"className":1842},[133],[24,1844,322],{"className":1845},[104,105],[24,1847],{"className":1848,"style":450},[110],[24,1850,1571],{"className":1851},[454],[24,1853],{"className":1854,"style":450},[110],[24,1856,1858,1861,1864,1867,1870,1873],{"className":1857},[95],[24,1859],{"className":1860,"style":125},[99],[24,1862,62],{"className":1863},[133],[24,1865,76],{"className":1866},[104],[24,1868],{"className":1869,"style":450},[110],[24,1871,380],{"className":1872},[454],[24,1874],{"className":1875,"style":450},[110],[24,1877,1879,1882,1885,1888,1918,1921,1924],{"className":1878},[95],[24,1880],{"className":1881,"style":125},[99],[24,1883,322],{"className":1884},[104,105],[24,1886,82],{"className":1887},[152],[24,1889,1891,1894],{"className":1890},[152],[24,1892,82],{"className":1893},[152],[24,1895,1897],{"className":1896},[700],[24,1898,1900],{"className":1899},[704],[24,1901,1903],{"className":1902},[709],[24,1904,1907],{"className":1905,"style":1906},[713],"height:0.7144em;",[24,1908,1909,1912],{"style":1047},[24,1910],{"className":1911,"style":722},[721],[24,1913,1915],{"className":1914},[726,727,728,729],[24,1916,584],{"className":1917},[104,105,729],[24,1919],{"className":1920,"style":111},[110],[24,1922,599],{"className":1923},[115],[24,1925],{"className":1926,"style":111},[110],[24,1928,1930,1933],{"className":1929},[95],[24,1931],{"className":1932,"style":296},[99],[24,1934,76],{"className":1935},[104],[20,1937,1938],{},"Thus",[24,1940,1942],{"className":1941,"translate":28},[27],[24,1943,1945,1974],{"className":1944,"translate":28},[32],[24,1946,1948],{"className":1947},[36],[38,1949,1950],{"xmlns":40,"display":41},[43,1951,1952,1972],{},[46,1953,1954,1960,1962,1964,1966,1968,1970],{},[577,1955,1956,1958],{},[49,1957,581],{},[49,1959,584],{},[53,1961,62],{"stretchy":61},[49,1963,51],{},[53,1965,82],{"stretchy":61},[53,1967,62],{"stretchy":61},[49,1969,322],{},[53,1971,82],{"stretchy":61},[84,1973,1416],{"encoding":86},[24,1975,1977],{"className":1976,"ariaHidden":72},[91],[24,1978,1980,1983,2023,2026,2029,2032,2035,2038],{"className":1979},[95],[24,1981],{"className":1982,"style":125},[99],[24,1984,1986,1989],{"className":1985},[104],[24,1987,581],{"className":1988,"style":696},[104,105],[24,1990,1992],{"className":1991},[700],[24,1993,1995,2015],{"className":1994},[704,705],[24,1996,1998,2012],{"className":1997},[709],[24,1999,2001],{"className":2000,"style":714},[713],[24,2002,2003,2006],{"style":717},[24,2004],{"className":2005,"style":722},[721],[24,2007,2009],{"className":2008},[726,727,728,729],[24,2010,584],{"className":2011},[104,105,729],[24,2013,737],{"className":2014},[736],[24,2016,2018],{"className":2017},[709],[24,2019,2021],{"className":2020,"style":744},[713],[24,2022],{},[24,2024,62],{"className":2025},[133],[24,2027,51],{"className":2028,"style":106},[104,105],[24,2030,82],{"className":2031},[152],[24,2033,62],{"className":2034},[133],[24,2036,322],{"className":2037},[104,105],[24,2039,82],{"className":2040},[152],[20,2042,2043],{},"is actually a weighted average of the sequence",[24,2045,2047],{"className":2046,"translate":28},[27],[24,2048,2050,2101],{"className":2049,"translate":28},[32],[24,2051,2053],{"className":2052},[36],[38,2054,2055],{"xmlns":40,"display":41},[43,2056,2057,2098],{},[46,2058,2059,2061,2063,2065,2067,2069,2071,2083,2085,2088,2090,2092,2094,2096],{},[49,2060,51],{},[53,2062,62],{"stretchy":61},[67,2064,69],{},[53,2066,82],{"stretchy":61},[53,2068,73],{"separator":72},[49,2070,51],{},[46,2072,2073,2075,2081],{},[53,2074,62],{"fence":72},[628,2076,2077,2079],{},[67,2078,76],{},[49,2080,584],{},[53,2082,82],{"fence":72},[53,2084,73],{"separator":72},[53,2086,2087],{},"…",[53,2089,73],{"separator":72},[49,2091,51],{},[53,2093,62],{"stretchy":61},[67,2095,76],{},[53,2097,82],{"stretchy":61},[84,2099,2100],{"encoding":86},"f(0),\nf\\left(\\frac1n\\right),\n\\dots,\nf(1)",[24,2102,2104],{"className":2103,"ariaHidden":72},[91],[24,2105,2107,2110,2113,2116,2119,2122,2125,2128,2131,2134,2212,2215,2218,2221,2224,2227,2230,2233,2236,2239,2242],{"className":2106},[95],[24,2108],{"className":2109,"style":1194},[99],[24,2111,51],{"className":2112,"style":106},[104,105],[24,2114,62],{"className":2115},[133],[24,2117,69],{"className":2118},[104],[24,2120,82],{"className":2121},[152],[24,2123,73],{"className":2124},[141],[24,2126],{"className":2127,"style":145},[110],[24,2129,51],{"className":2130,"style":106},[104,105],[24,2132],{"className":2133,"style":145},[110],[24,2135,2137,2143,2206],{"className":2136},[875],[24,2138,2140],{"className":2139,"style":880},[133,879],[24,2141,62],{"className":2142},[884,728],[24,2144,2146,2149,2203],{"className":2145},[104],[24,2147],{"className":2148},[133,891],[24,2150,2152],{"className":2151},[628],[24,2153,2155,2195],{"className":2154},[704,705],[24,2156,2158,2192],{"className":2157},[709],[24,2159,2162,2173,2181],{"className":2160,"style":2161},[713],"height:1.3214em;",[24,2163,2164,2167],{"style":907},[24,2165],{"className":2166,"style":911},[721],[24,2168,2170],{"className":2169},[104],[24,2171,584],{"className":2172},[104,105],[24,2174,2175,2178],{"style":920},[24,2176],{"className":2177,"style":911},[721],[24,2179],{"className":2180,"style":928},[927],[24,2182,2183,2186],{"style":931},[24,2184],{"className":2185,"style":911},[721],[24,2187,2189],{"className":2188},[104],[24,2190,76],{"className":2191},[104],[24,2193,737],{"className":2194},[736],[24,2196,2198],{"className":2197},[709],[24,2199,2201],{"className":2200,"style":950},[713],[24,2202],{},[24,2204],{"className":2205},[152,891],[24,2207,2209],{"className":2208,"style":880},[152,879],[24,2210,82],{"className":2211},[884,728],[24,2213],{"className":2214,"style":145},[110],[24,2216,73],{"className":2217},[141],[24,2219],{"className":2220,"style":145},[110],[24,2222,2087],{"className":2223},[875],[24,2225],{"className":2226,"style":145},[110],[24,2228,73],{"className":2229},[141],[24,2231],{"className":2232,"style":145},[110],[24,2234,51],{"className":2235,"style":106},[104,105],[24,2237,62],{"className":2238},[133],[24,2240,76],{"className":2241},[104],[24,2243,82],{"className":2244},[152],[20,2246,2247],{},"where the weights are",[24,2249,2251],{"className":2250,"translate":28},[27],[24,2252,2254,2326],{"className":2253,"translate":28},[32],[24,2255,2257],{"className":2256},[36],[38,2258,2259],{"xmlns":40,"display":41},[43,2260,2261,2323],{},[46,2262,2263,2275,2277,2279,2281,2283,2295,2301,2303,2305,2307,2309,2321],{},[577,2264,2265,2267],{},[49,2266,20],{},[46,2268,2269,2271,2273],{},[49,2270,584],{},[53,2272,73],{"separator":72},[49,2274,610],{},[53,2276,62],{"stretchy":61},[49,2278,322],{},[53,2280,82],{"stretchy":61},[53,2282,599],{},[46,2284,2285,2287,2293],{},[53,2286,62],{"fence":72},[628,2288,2289,2291],{"linethickness":642},[49,2290,584],{},[49,2292,610],{},[53,2294,82],{"fence":72},[650,2296,2297,2299],{},[49,2298,322],{},[49,2300,610],{},[53,2302,62],{"stretchy":61},[67,2304,76],{},[53,2306,380],{},[49,2308,322],{},[650,2310,2311,2313],{},[53,2312,82],{"stretchy":61},[46,2314,2315,2317,2319],{},[49,2316,584],{},[53,2318,380],{},[49,2320,610],{},[49,2322,238],{"mathvariant":368},[84,2324,2325],{"encoding":86},"p_{n,k}(x)\n=\n\\binom{n}{k}\nx^k(1-x)^{n-k}.",[24,2327,2329,2406,2516],{"className":2328,"ariaHidden":72},[91],[24,2330,2332,2336,2388,2391,2394,2397,2400,2403],{"className":2331},[95],[24,2333],{"className":2334,"style":2335},[99],"height:1.0361em;vertical-align:-0.2861em;",[24,2337,2339,2342],{"className":2338},[104],[24,2340,20],{"className":2341},[104,105],[24,2343,2345],{"className":2344},[700],[24,2346,2348,2379],{"className":2347},[704,705],[24,2349,2351,2376],{"className":2350},[709],[24,2352,2355],{"className":2353,"style":2354},[713],"height:0.3361em;",[24,2356,2358,2361],{"style":2357},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[24,2359],{"className":2360,"style":722},[721],[24,2362,2364],{"className":2363},[726,727,728,729],[24,2365,2367,2370,2373],{"className":2366},[104,729],[24,2368,584],{"className":2369},[104,105,729],[24,2371,73],{"className":2372},[141,729],[24,2374,610],{"className":2375,"style":812},[104,105,729],[24,2377,737],{"className":2378},[736],[24,2380,2382],{"className":2381},[709],[24,2383,2386],{"className":2384,"style":2385},[713],"height:0.2861em;",[24,2387],{},[24,2389,62],{"className":2390},[133],[24,2392,322],{"className":2393},[104,105],[24,2395,82],{"className":2396},[152],[24,2398],{"className":2399,"style":111},[110],[24,2401,599],{"className":2402},[115],[24,2404],{"className":2405,"style":111},[110],[24,2407,2409,2412,2472,2501,2504,2507,2510,2513],{"className":2408},[95],[24,2410],{"className":2411,"style":1194},[99],[24,2413,2415,2421,2466],{"className":2414},[104],[24,2416,2418],{"className":2417,"style":880},[133,879],[24,2419,62],{"className":2420},[884,728],[24,2422,2424],{"className":2423},[628],[24,2425,2427,2458],{"className":2426},[704,705],[24,2428,2430,2455],{"className":2429},[709],[24,2431,2433,2444],{"className":2432,"style":986},[713],[24,2434,2435,2438],{"style":907},[24,2436],{"className":2437,"style":911},[721],[24,2439,2441],{"className":2440},[104],[24,2442,610],{"className":2443,"style":812},[104,105],[24,2445,2446,2449],{"style":931},[24,2447],{"className":2448,"style":911},[721],[24,2450,2452],{"className":2451},[104],[24,2453,584],{"className":2454},[104,105],[24,2456,737],{"className":2457},[736],[24,2459,2461],{"className":2460},[709],[24,2462,2464],{"className":2463,"style":950},[713],[24,2465],{},[24,2467,2469],{"className":2468,"style":880},[152,879],[24,2470,82],{"className":2471},[884,728],[24,2473,2475,2478],{"className":2474},[104],[24,2476,322],{"className":2477},[104,105],[24,2479,2481],{"className":2480},[700],[24,2482,2484],{"className":2483},[704],[24,2485,2487],{"className":2486},[709],[24,2488,2490],{"className":2489,"style":1044},[713],[24,2491,2492,2495],{"style":1047},[24,2493],{"className":2494,"style":722},[721],[24,2496,2498],{"className":2497},[726,727,728,729],[24,2499,610],{"className":2500,"style":812},[104,105,729],[24,2502,62],{"className":2503},[133],[24,2505,76],{"className":2506},[104],[24,2508],{"className":2509,"style":450},[110],[24,2511,380],{"className":2512},[454],[24,2514],{"className":2515,"style":450},[110],[24,2517,2519,2522,2525,2563],{"className":2518},[95],[24,2520],{"className":2521,"style":1078},[99],[24,2523,322],{"className":2524},[104,105],[24,2526,2528,2531],{"className":2527},[152],[24,2529,82],{"className":2530},[152],[24,2532,2534],{"className":2533},[700],[24,2535,2537],{"className":2536},[704],[24,2538,2540],{"className":2539},[709],[24,2541,2543],{"className":2542,"style":1044},[713],[24,2544,2545,2548],{"style":1047},[24,2546],{"className":2547,"style":722},[721],[24,2549,2551],{"className":2550},[726,727,728,729],[24,2552,2554,2557,2560],{"className":2553},[104,729],[24,2555,584],{"className":2556},[104,105,729],[24,2558,380],{"className":2559},[454,729],[24,2561,610],{"className":2562,"style":812},[104,105,729],[24,2564,238],{"className":2565},[104],[20,2567,2568],{},"And the sum of these weights is 1.",[20,2570,2571],{},"But we can also interpret it from a probabilistic perspective.",[20,2573,2574],{},"Let the random variable",[24,2576,2578],{"className":2577,"translate":28},[27],[24,2579,2581,2638],{"className":2580,"translate":28},[32],[24,2582,2584],{"className":2583},[36],[38,2585,2586],{"xmlns":40,"display":41},[43,2587,2588,2635],{},[46,2589,2590,2597,2600,2623,2625,2627,2629,2631,2633],{},[577,2591,2592,2595],{},[49,2593,2594],{},"X",[49,2596,584],{},[53,2598,2599],{},"∼",[46,2601,2602,2604,2607,2609,2612,2615,2617,2620],{},[49,2603,581],{"mathvariant":368},[49,2605,2606],{"mathvariant":368},"i",[49,2608,584],{"mathvariant":368},[49,2610,2611],{"mathvariant":368},"o",[49,2613,2614],{"mathvariant":368},"m",[49,2616,2606],{"mathvariant":368},[49,2618,2619],{"mathvariant":368},"a",[49,2621,2622],{"mathvariant":368},"l",[53,2624,62],{"stretchy":61},[49,2626,584],{},[53,2628,73],{"separator":72},[49,2630,322],{},[53,2632,82],{"stretchy":61},[49,2634,238],{"mathvariant":368},[84,2636,2637],{"encoding":86},"X_n\\sim \\mathrm{Binomial}(n,x).",[24,2639,2641,2699],{"className":2640,"ariaHidden":72},[91],[24,2642,2644,2648,2690,2693,2696],{"className":2643},[95],[24,2645],{"className":2646,"style":2647},[99],"height:0.8333em;vertical-align:-0.15em;",[24,2649,2651,2655],{"className":2650},[104],[24,2652,2594],{"className":2653,"style":2654},[104,105],"margin-right:0.0785em;",[24,2656,2658],{"className":2657},[700],[24,2659,2661,2682],{"className":2660},[704,705],[24,2662,2664,2679],{"className":2663},[709],[24,2665,2667],{"className":2666,"style":714},[713],[24,2668,2670,2673],{"style":2669},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[24,2671],{"className":2672,"style":722},[721],[24,2674,2676],{"className":2675},[726,727,728,729],[24,2677,584],{"className":2678},[104,105,729],[24,2680,737],{"className":2681},[736],[24,2683,2685],{"className":2684},[709],[24,2686,2688],{"className":2687,"style":744},[713],[24,2689],{},[24,2691],{"className":2692,"style":111},[110],[24,2694,2599],{"className":2695},[115],[24,2697],{"className":2698,"style":111},[110],[24,2700,2702,2705,2713,2716,2719,2722,2725,2728,2731],{"className":2701},[95],[24,2703],{"className":2704,"style":125},[99],[24,2706,2708],{"className":2707},[104],[24,2709,2712],{"className":2710},[104,2711],"mathrm","Binomial",[24,2714,62],{"className":2715},[133],[24,2717,584],{"className":2718},[104,105],[24,2720,73],{"className":2721},[141],[24,2723],{"className":2724,"style":145},[110],[24,2726,322],{"className":2727},[104,105],[24,2729,82],{"className":2730},[152],[24,2732,238],{"className":2733},[104],[20,2735,2736],{},"That is,",[24,2738,2740],{"className":2739,"translate":28},[27],[24,2741,2743,2813],{"className":2742,"translate":28},[32],[24,2744,2746],{"className":2745},[36],[38,2747,2748],{"xmlns":40,"display":41},[43,2749,2750,2810],{},[46,2751,2752,2754,2756,2762,2764,2766,2768,2770,2782,2788,2790,2792,2794,2796,2808],{},[49,2753,317],{},[53,2755,62],{"stretchy":61},[577,2757,2758,2760],{},[49,2759,2594],{},[49,2761,584],{},[53,2763,599],{},[49,2765,610],{},[53,2767,82],{"stretchy":61},[53,2769,599],{},[46,2771,2772,2774,2780],{},[53,2773,62],{"fence":72},[628,2775,2776,2778],{"linethickness":642},[49,2777,584],{},[49,2779,610],{},[53,2781,82],{"fence":72},[650,2783,2784,2786],{},[49,2785,322],{},[49,2787,610],{},[53,2789,62],{"stretchy":61},[67,2791,76],{},[53,2793,380],{},[49,2795,322],{},[650,2797,2798,2800],{},[53,2799,82],{"stretchy":61},[46,2801,2802,2804,2806],{},[49,2803,584],{},[53,2805,380],{},[49,2807,610],{},[49,2809,238],{"mathvariant":368},[84,2811,2812],{"encoding":86},"P(X_n=k)\n=\n\\binom{n}{k}\nx^k(1-x)^{n-k}.",[24,2814,2816,2877,2898,3008],{"className":2815,"ariaHidden":72},[91],[24,2817,2819,2822,2825,2828,2868,2871,2874],{"className":2818},[95],[24,2820],{"className":2821,"style":125},[99],[24,2823,317],{"className":2824,"style":340},[104,105],[24,2826,62],{"className":2827},[133],[24,2829,2831,2834],{"className":2830},[104],[24,2832,2594],{"className":2833,"style":2654},[104,105],[24,2835,2837],{"className":2836},[700],[24,2838,2840,2860],{"className":2839},[704,705],[24,2841,2843,2857],{"className":2842},[709],[24,2844,2846],{"className":2845,"style":714},[713],[24,2847,2848,2851],{"style":2669},[24,2849],{"className":2850,"style":722},[721],[24,2852,2854],{"className":2853},[726,727,728,729],[24,2855,584],{"className":2856},[104,105,729],[24,2858,737],{"className":2859},[736],[24,2861,2863],{"className":2862},[709],[24,2864,2866],{"className":2865,"style":744},[713],[24,2867],{},[24,2869],{"className":2870,"style":111},[110],[24,2872,599],{"className":2873},[115],[24,2875],{"className":2876,"style":111},[110],[24,2878,2880,2883,2886,2889,2892,2895],{"className":2879},[95],[24,2881],{"className":2882,"style":125},[99],[24,2884,610],{"className":2885,"style":812},[104,105],[24,2887,82],{"className":2888},[152],[24,2890],{"className":2891,"style":111},[110],[24,2893,599],{"className":2894},[115],[24,2896],{"className":2897,"style":111},[110],[24,2899,2901,2904,2964,2993,2996,2999,3002,3005],{"className":2900},[95],[24,2902],{"className":2903,"style":1194},[99],[24,2905,2907,2913,2958],{"className":2906},[104],[24,2908,2910],{"className":2909,"style":880},[133,879],[24,2911,62],{"className":2912},[884,728],[24,2914,2916],{"className":2915},[628],[24,2917,2919,2950],{"className":2918},[704,705],[24,2920,2922,2947],{"className":2921},[709],[24,2923,2925,2936],{"className":2924,"style":986},[713],[24,2926,2927,2930],{"style":907},[24,2928],{"className":2929,"style":911},[721],[24,2931,2933],{"className":2932},[104],[24,2934,610],{"className":2935,"style":812},[104,105],[24,2937,2938,2941],{"style":931},[24,2939],{"className":2940,"style":911},[721],[24,2942,2944],{"className":2943},[104],[24,2945,584],{"className":2946},[104,105],[24,2948,737],{"className":2949},[736],[24,2951,2953],{"className":2952},[709],[24,2954,2956],{"className":2955,"style":950},[713],[24,2957],{},[24,2959,2961],{"className":2960,"style":880},[152,879],[24,2962,82],{"className":2963},[884,728],[24,2965,2967,2970],{"className":2966},[104],[24,2968,322],{"className":2969},[104,105],[24,2971,2973],{"className":2972},[700],[24,2974,2976],{"className":2975},[704],[24,2977,2979],{"className":2978},[709],[24,2980,2982],{"className":2981,"style":1044},[713],[24,2983,2984,2987],{"style":1047},[24,2985],{"className":2986,"style":722},[721],[24,2988,2990],{"className":2989},[726,727,728,729],[24,2991,610],{"className":2992,"style":812},[104,105,729],[24,2994,62],{"className":2995},[133],[24,2997,76],{"className":2998},[104],[24,3000],{"className":3001,"style":450},[110],[24,3003,380],{"className":3004},[454],[24,3006],{"className":3007,"style":450},[110],[24,3009,3011,3014,3017,3055],{"className":3010},[95],[24,3012],{"className":3013,"style":1078},[99],[24,3015,322],{"className":3016},[104,105],[24,3018,3020,3023],{"className":3019},[152],[24,3021,82],{"className":3022},[152],[24,3024,3026],{"className":3025},[700],[24,3027,3029],{"className":3028},[704],[24,3030,3032],{"className":3031},[709],[24,3033,3035],{"className":3034,"style":1044},[713],[24,3036,3037,3040],{"style":1047},[24,3038],{"className":3039,"style":722},[721],[24,3041,3043],{"className":3042},[726,727,728,729],[24,3044,3046,3049,3052],{"className":3045},[104,729],[24,3047,584],{"className":3048},[104,105,729],[24,3050,380],{"className":3051},[454,729],[24,3053,610],{"className":3054,"style":812},[104,105,729],[24,3056,238],{"className":3057},[104],[20,3059,3060],{},"Then",[24,3062,3064],{"className":3063,"translate":28},[27],[24,3065,3067,3132],{"className":3066,"translate":28},[32],[24,3068,3070],{"className":3069},[36],[38,3071,3072],{"xmlns":40,"display":41},[43,3073,3074,3129],{},[46,3075,3076,3082,3084,3086,3088,3090,3092,3094,3096,3099,3101,3127],{},[577,3077,3078,3080],{},[49,3079,581],{},[49,3081,584],{},[53,3083,62],{"stretchy":61},[49,3085,51],{},[53,3087,82],{"stretchy":61},[53,3089,62],{"stretchy":61},[49,3091,322],{},[53,3093,82],{"stretchy":61},[53,3095,599],{},[49,3097,3098],{},"E",[620,3100,622],{},[46,3102,3103,3105,3107,3109,3125],{},[53,3104,65],{"fence":72},[49,3106,51],{},[620,3108,622],{},[46,3110,3111,3113,3123],{},[53,3112,62],{"fence":72},[628,3114,3115,3121],{},[577,3116,3117,3119],{},[49,3118,2594],{},[49,3120,584],{},[49,3122,584],{},[53,3124,82],{"fence":72},[53,3126,79],{"fence":72},[49,3128,238],{"mathvariant":368},[84,3130,3131],{"encoding":86},"B_n(f)(x)\n=\n\nE\\!\\left[\nf\\!\\left(\\frac{X_n}{n}\\right)\n\\right].",[24,3133,3135,3208],{"className":3134,"ariaHidden":72},[91],[24,3136,3138,3141,3181,3184,3187,3190,3193,3196,3199,3202,3205],{"className":3137},[95],[24,3139],{"className":3140,"style":125},[99],[24,3142,3144,3147],{"className":3143},[104],[24,3145,581],{"className":3146,"style":696},[104,105],[24,3148,3150],{"className":3149},[700],[24,3151,3153,3173],{"className":3152},[704,705],[24,3154,3156,3170],{"className":3155},[709],[24,3157,3159],{"className":3158,"style":714},[713],[24,3160,3161,3164],{"style":717},[24,3162],{"className":3163,"style":722},[721],[24,3165,3167],{"className":3166},[726,727,728,729],[24,3168,584],{"className":3169},[104,105,729],[24,3171,737],{"className":3172},[736],[24,3174,3176],{"className":3175},[709],[24,3177,3179],{"className":3178,"style":744},[713],[24,3180],{},[24,3182,62],{"className":3183},[133],[24,3185,51],{"className":3186,"style":106},[104,105],[24,3188,82],{"className":3189},[152],[24,3191,62],{"className":3192},[133],[24,3194,322],{"className":3195},[104,105],[24,3197,82],{"className":3198},[152],[24,3200],{"className":3201,"style":111},[110],[24,3203,599],{"className":3204},[115],[24,3206],{"className":3207,"style":111},[110],[24,3209,3211,3214,3218,3221,3224,3363,3366],{"className":3210},[95],[24,3212],{"className":3213,"style":1194},[99],[24,3215,3098],{"className":3216,"style":3217},[104,105],"margin-right:0.0576em;",[24,3219],{"className":3220,"style":868},[110],[24,3222],{"className":3223,"style":145},[110],[24,3225,3227,3233,3236,3239,3242,3357],{"className":3226},[875],[24,3228,3230],{"className":3229,"style":880},[133,879],[24,3231,65],{"className":3232},[884,728],[24,3234,51],{"className":3235,"style":106},[104,105],[24,3237],{"className":3238,"style":868},[110],[24,3240],{"className":3241,"style":145},[110],[24,3243,3245,3251,3351],{"className":3244},[875],[24,3246,3248],{"className":3247,"style":880},[133,879],[24,3249,62],{"className":3250},[884,728],[24,3252,3254,3257,3348],{"className":3253},[104],[24,3255],{"className":3256},[133,891],[24,3258,3260],{"className":3259},[628],[24,3261,3263,3340],{"className":3262},[704,705],[24,3264,3266,3337],{"className":3265},[709],[24,3267,3270,3281,3289],{"className":3268,"style":3269},[713],"height:1.3603em;",[24,3271,3272,3275],{"style":907},[24,3273],{"className":3274,"style":911},[721],[24,3276,3278],{"className":3277},[104],[24,3279,584],{"className":3280},[104,105],[24,3282,3283,3286],{"style":920},[24,3284],{"className":3285,"style":911},[721],[24,3287],{"className":3288,"style":928},[927],[24,3290,3291,3294],{"style":931},[24,3292],{"className":3293,"style":911},[721],[24,3295,3297],{"className":3296},[104],[24,3298,3300,3303],{"className":3299},[104],[24,3301,2594],{"className":3302,"style":2654},[104,105],[24,3304,3306],{"className":3305},[700],[24,3307,3309,3329],{"className":3308},[704,705],[24,3310,3312,3326],{"className":3311},[709],[24,3313,3315],{"className":3314,"style":714},[713],[24,3316,3317,3320],{"style":2669},[24,3318],{"className":3319,"style":722},[721],[24,3321,3323],{"className":3322},[726,727,728,729],[24,3324,584],{"className":3325},[104,105,729],[24,3327,737],{"className":3328},[736],[24,3330,3332],{"className":3331},[709],[24,3333,3335],{"className":3334,"style":744},[713],[24,3336],{},[24,3338,737],{"className":3339},[736],[24,3341,3343],{"className":3342},[709],[24,3344,3346],{"className":3345,"style":950},[713],[24,3347],{},[24,3349],{"className":3350},[152,891],[24,3352,3354],{"className":3353,"style":880},[152,879],[24,3355,82],{"className":3356},[884,728],[24,3358,3360],{"className":3359,"style":880},[152,879],[24,3361,79],{"className":3362},[884,728],[24,3364],{"className":3365,"style":145},[110],[24,3367,238],{"className":3368},[104],[20,3370,3371],{},"This step is very elegant. The Bernstein polynomial is actually the expectation of",[24,3373,3375],{"className":3374,"translate":28},[27],[24,3376,3378,3410],{"className":3377,"translate":28},[32],[24,3379,3381],{"className":3380},[36],[38,3382,3383],{"xmlns":40,"display":41},[43,3384,3385,3407],{},[46,3386,3387,3389,3405],{},[49,3388,51],{},[46,3390,3391,3393,3403],{},[53,3392,62],{"fence":72},[628,3394,3395,3401],{},[577,3396,3397,3399],{},[49,3398,2594],{},[49,3400,584],{},[49,3402,584],{},[53,3404,82],{"fence":72},[49,3406,238],{"mathvariant":368},[84,3408,3409],{"encoding":86},"f\\left(\\frac{X_n}{n}\\right).",[24,3411,3413],{"className":3412,"ariaHidden":72},[91],[24,3414,3416,3419,3422,3425,3539,3542],{"className":3415},[95],[24,3417],{"className":3418,"style":1194},[99],[24,3420,51],{"className":3421,"style":106},[104,105],[24,3423],{"className":3424,"style":145},[110],[24,3426,3428,3434,3533],{"className":3427},[875],[24,3429,3431],{"className":3430,"style":880},[133,879],[24,3432,62],{"className":3433},[884,728],[24,3435,3437,3440,3530],{"className":3436},[104],[24,3438],{"className":3439},[133,891],[24,3441,3443],{"className":3442},[628],[24,3444,3446,3522],{"className":3445},[704,705],[24,3447,3449,3519],{"className":3448},[709],[24,3450,3452,3463,3471],{"className":3451,"style":3269},[713],[24,3453,3454,3457],{"style":907},[24,3455],{"className":3456,"style":911},[721],[24,3458,3460],{"className":3459},[104],[24,3461,584],{"className":3462},[104,105],[24,3464,3465,3468],{"style":920},[24,3466],{"className":3467,"style":911},[721],[24,3469],{"className":3470,"style":928},[927],[24,3472,3473,3476],{"style":931},[24,3474],{"className":3475,"style":911},[721],[24,3477,3479],{"className":3478},[104],[24,3480,3482,3485],{"className":3481},[104],[24,3483,2594],{"className":3484,"style":2654},[104,105],[24,3486,3488],{"className":3487},[700],[24,3489,3491,3511],{"className":3490},[704,705],[24,3492,3494,3508],{"className":3493},[709],[24,3495,3497],{"className":3496,"style":714},[713],[24,3498,3499,3502],{"style":2669},[24,3500],{"className":3501,"style":722},[721],[24,3503,3505],{"className":3504},[726,727,728,729],[24,3506,584],{"className":3507},[104,105,729],[24,3509,737],{"className":3510},[736],[24,3512,3514],{"className":3513},[709],[24,3515,3517],{"className":3516,"style":744},[713],[24,3518],{},[24,3520,737],{"className":3521},[736],[24,3523,3525],{"className":3524},[709],[24,3526,3528],{"className":3527,"style":950},[713],[24,3529],{},[24,3531],{"className":3532},[152,891],[24,3534,3536],{"className":3535,"style":880},[152,879],[24,3537,82],{"className":3538},[884,728],[24,3540],{"className":3541,"style":145},[110],[24,3543,238],{"className":3544},[104],[20,3546,3547,3548,238],{},"Now let's analyze why it approximates ",[24,3549,3551,3571],{"className":3550,"translate":28},[32],[24,3552,3554],{"className":3553},[36],[38,3555,3556],{"xmlns":40},[43,3557,3558,3568],{},[46,3559,3560,3562,3564,3566],{},[49,3561,51],{},[53,3563,62],{"stretchy":61},[49,3565,322],{},[53,3567,82],{"stretchy":61},[84,3569,3570],{"encoding":86},"f(x)",[24,3572,3574],{"className":3573,"ariaHidden":72},[91],[24,3575,3577,3580,3583,3586,3589],{"className":3576},[95],[24,3578],{"className":3579,"style":125},[99],[24,3581,51],{"className":3582,"style":106},[104,105],[24,3584,62],{"className":3585},[133],[24,3587,322],{"className":3588},[104,105],[24,3590,82],{"className":3591},[152],[20,3593,3594],{},"For the binomial distribution we have",[24,3596,3598],{"className":3597,"translate":28},[27],[24,3599,3601,3639],{"className":3600,"translate":28},[32],[24,3602,3604],{"className":3603},[36],[38,3605,3606],{"xmlns":40,"display":41},[43,3607,3608,3636],{},[46,3609,3610,3612,3614,3630,3632,3634],{},[49,3611,3098],{},[620,3613,622],{},[46,3615,3616,3618,3628],{},[53,3617,62],{"fence":72},[628,3619,3620,3626],{},[577,3621,3622,3624],{},[49,3623,2594],{},[49,3625,584],{},[49,3627,584],{},[53,3629,82],{"fence":72},[53,3631,599],{},[49,3633,322],{},[53,3635,73],{"separator":72},[84,3637,3638],{"encoding":86},"E\\!\\left(\\frac{X_n}{n}\\right)\n=\nx,",[24,3640,3642,3780],{"className":3641,"ariaHidden":72},[91],[24,3643,3645,3648,3651,3654,3657,3771,3774,3777],{"className":3644},[95],[24,3646],{"className":3647,"style":1194},[99],[24,3649,3098],{"className":3650,"style":3217},[104,105],[24,3652],{"className":3653,"style":868},[110],[24,3655],{"className":3656,"style":145},[110],[24,3658,3660,3666,3765],{"className":3659},[875],[24,3661,3663],{"className":3662,"style":880},[133,879],[24,3664,62],{"className":3665},[884,728],[24,3667,3669,3672,3762],{"className":3668},[104],[24,3670],{"className":3671},[133,891],[24,3673,3675],{"className":3674},[628],[24,3676,3678,3754],{"className":3677},[704,705],[24,3679,3681,3751],{"className":3680},[709],[24,3682,3684,3695,3703],{"className":3683,"style":3269},[713],[24,3685,3686,3689],{"style":907},[24,3687],{"className":3688,"style":911},[721],[24,3690,3692],{"className":3691},[104],[24,3693,584],{"className":3694},[104,105],[24,3696,3697,3700],{"style":920},[24,3698],{"className":3699,"style":911},[721],[24,3701],{"className":3702,"style":928},[927],[24,3704,3705,3708],{"style":931},[24,3706],{"className":3707,"style":911},[721],[24,3709,3711],{"className":3710},[104],[24,3712,3714,3717],{"className":3713},[104],[24,3715,2594],{"className":3716,"style":2654},[104,105],[24,3718,3720],{"className":3719},[700],[24,3721,3723,3743],{"className":3722},[704,705],[24,3724,3726,3740],{"className":3725},[709],[24,3727,3729],{"className":3728,"style":714},[713],[24,3730,3731,3734],{"style":2669},[24,3732],{"className":3733,"style":722},[721],[24,3735,3737],{"className":3736},[726,727,728,729],[24,3738,584],{"className":3739},[104,105,729],[24,3741,737],{"className":3742},[736],[24,3744,3746],{"className":3745},[709],[24,3747,3749],{"className":3748,"style":744},[713],[24,3750],{},[24,3752,737],{"className":3753},[736],[24,3755,3757],{"className":3756},[709],[24,3758,3760],{"className":3759,"style":950},[713],[24,3761],{},[24,3763],{"className":3764},[152,891],[24,3766,3768],{"className":3767,"style":880},[152,879],[24,3769,82],{"className":3770},[884,728],[24,3772],{"className":3773,"style":111},[110],[24,3775,599],{"className":3776},[115],[24,3778],{"className":3779,"style":111},[110],[24,3781,3783,3787,3790],{"className":3782},[95],[24,3784],{"className":3785,"style":3786},[99],"height:0.625em;vertical-align:-0.1944em;",[24,3788,322],{"className":3789},[104,105],[24,3791,73],{"className":3792},[141],[24,3794,3796],{"className":3795,"translate":28},[27],[24,3797,3799,3861],{"className":3798,"translate":28},[32],[24,3800,3802],{"className":3801},[36],[38,3803,3804],{"xmlns":40,"display":41},[43,3805,3806,3858],{},[46,3807,3808,3818,3820,3836,3838,3856],{},[46,3809,3810,3813,3815],{},[49,3811,3812],{"mathvariant":368},"V",[49,3814,2619],{"mathvariant":368},[49,3816,3817],{"mathvariant":368},"r",[620,3819,622],{},[46,3821,3822,3824,3834],{},[53,3823,62],{"fence":72},[628,3825,3826,3832],{},[577,3827,3828,3830],{},[49,3829,2594],{},[49,3831,584],{},[49,3833,584],{},[53,3835,82],{"fence":72},[53,3837,599],{},[628,3839,3840,3854],{},[46,3841,3842,3844,3846,3848,3850,3852],{},[49,3843,322],{},[53,3845,62],{"stretchy":61},[67,3847,76],{},[53,3849,380],{},[49,3851,322],{},[53,3853,82],{"stretchy":61},[49,3855,584],{},[49,3857,238],{"mathvariant":368},[84,3859,3860],{"encoding":86},"\\mathrm{Var}\n\\!\\left(\n\\frac{X_n}{n}\n\\right)\n=\n\\frac{x(1-x)}{n}.",[24,3862,3864,4006],{"className":3863,"ariaHidden":72},[91],[24,3865,3867,3870,3877,3880,3883,3997,4000,4003],{"className":3866},[95],[24,3868],{"className":3869,"style":1194},[99],[24,3871,3873],{"className":3872},[104],[24,3874,3876],{"className":3875},[104,2711],"Var",[24,3878],{"className":3879,"style":868},[110],[24,3881],{"className":3882,"style":145},[110],[24,3884,3886,3892,3991],{"className":3885},[875],[24,3887,3889],{"className":3888,"style":880},[133,879],[24,3890,62],{"className":3891},[884,728],[24,3893,3895,3898,3988],{"className":3894},[104],[24,3896],{"className":3897},[133,891],[24,3899,3901],{"className":3900},[628],[24,3902,3904,3980],{"className":3903},[704,705],[24,3905,3907,3977],{"className":3906},[709],[24,3908,3910,3921,3929],{"className":3909,"style":3269},[713],[24,3911,3912,3915],{"style":907},[24,3913],{"className":3914,"style":911},[721],[24,3916,3918],{"className":3917},[104],[24,3919,584],{"className":3920},[104,105],[24,3922,3923,3926],{"style":920},[24,3924],{"className":3925,"style":911},[721],[24,3927],{"className":3928,"style":928},[927],[24,3930,3931,3934],{"style":931},[24,3932],{"className":3933,"style":911},[721],[24,3935,3937],{"className":3936},[104],[24,3938,3940,3943],{"className":3939},[104],[24,3941,2594],{"className":3942,"style":2654},[104,105],[24,3944,3946],{"className":3945},[700],[24,3947,3949,3969],{"className":3948},[704,705],[24,3950,3952,3966],{"className":3951},[709],[24,3953,3955],{"className":3954,"style":714},[713],[24,3956,3957,3960],{"style":2669},[24,3958],{"className":3959,"style":722},[721],[24,3961,3963],{"className":3962},[726,727,728,729],[24,3964,584],{"className":3965},[104,105,729],[24,3967,737],{"className":3968},[736],[24,3970,3972],{"className":3971},[709],[24,3973,3975],{"className":3974,"style":744},[713],[24,3976],{},[24,3978,737],{"className":3979},[736],[24,3981,3983],{"className":3982},[709],[24,3984,3986],{"className":3985,"style":950},[713],[24,3987],{},[24,3989],{"className":3990},[152,891],[24,3992,3994],{"className":3993,"style":880},[152,879],[24,3995,82],{"className":3996},[884,728],[24,3998],{"className":3999,"style":111},[110],[24,4001,599],{"className":4002},[115],[24,4004],{"className":4005,"style":111},[110],[24,4007,4009,4013,4097],{"className":4008},[95],[24,4010],{"className":4011,"style":4012},[99],"height:2.113em;vertical-align:-0.686em;",[24,4014,4016,4019,4094],{"className":4015},[104],[24,4017],{"className":4018},[133,891],[24,4020,4022],{"className":4021},[628],[24,4023,4025,4086],{"className":4024},[704,705],[24,4026,4028,4083],{"className":4027},[709],[24,4029,4032,4043,4051],{"className":4030,"style":4031},[713],"height:1.427em;",[24,4033,4034,4037],{"style":907},[24,4035],{"className":4036,"style":911},[721],[24,4038,4040],{"className":4039},[104],[24,4041,584],{"className":4042},[104,105],[24,4044,4045,4048],{"style":920},[24,4046],{"className":4047,"style":911},[721],[24,4049],{"className":4050,"style":928},[927],[24,4052,4053,4056],{"style":931},[24,4054],{"className":4055,"style":911},[721],[24,4057,4059,4062,4065,4068,4071,4074,4077,4080],{"className":4058},[104],[24,4060,322],{"className":4061},[104,105],[24,4063,62],{"className":4064},[133],[24,4066,76],{"className":4067},[104],[24,4069],{"className":4070,"style":450},[110],[24,4072,380],{"className":4073},[454],[24,4075],{"className":4076,"style":450},[110],[24,4078,322],{"className":4079},[104,105],[24,4081,82],{"className":4082},[152],[24,4084,737],{"className":4085},[736],[24,4087,4089],{"className":4088},[709],[24,4090,4092],{"className":4091,"style":950},[713],[24,4093],{},[24,4095],{"className":4096},[152,891],[24,4098,238],{"className":4099},[104],[20,4101,1380],{},[24,4103,4105],{"className":4104,"translate":28},[27],[24,4106,4108,4130],{"className":4107,"translate":28},[32],[24,4109,4111],{"className":4110},[36],[38,4112,4113],{"xmlns":40,"display":41},[43,4114,4115,4127],{},[46,4116,4117],{},[628,4118,4119,4125],{},[577,4120,4121,4123],{},[49,4122,2594],{},[49,4124,584],{},[49,4126,584],{},[84,4128,4129],{"encoding":86},"\\frac{X_n}{n}",[24,4131,4133],{"className":4132,"ariaHidden":72},[91],[24,4134,4136,4140],{"className":4135},[95],[24,4137],{"className":4138,"style":4139},[99],"height:2.0463em;vertical-align:-0.686em;",[24,4141,4143,4146,4236],{"className":4142},[104],[24,4144],{"className":4145},[133,891],[24,4147,4149],{"className":4148},[628],[24,4150,4152,4228],{"className":4151},[704,705],[24,4153,4155,4225],{"className":4154},[709],[24,4156,4158,4169,4177],{"className":4157,"style":3269},[713],[24,4159,4160,4163],{"style":907},[24,4161],{"className":4162,"style":911},[721],[24,4164,4166],{"className":4165},[104],[24,4167,584],{"className":4168},[104,105],[24,4170,4171,4174],{"style":920},[24,4172],{"className":4173,"style":911},[721],[24,4175],{"className":4176,"style":928},[927],[24,4178,4179,4182],{"style":931},[24,4180],{"className":4181,"style":911},[721],[24,4183,4185],{"className":4184},[104],[24,4186,4188,4191],{"className":4187},[104],[24,4189,2594],{"className":4190,"style":2654},[104,105],[24,4192,4194],{"className":4193},[700],[24,4195,4197,4217],{"className":4196},[704,705],[24,4198,4200,4214],{"className":4199},[709],[24,4201,4203],{"className":4202,"style":714},[713],[24,4204,4205,4208],{"style":2669},[24,4206],{"className":4207,"style":722},[721],[24,4209,4211],{"className":4210},[726,727,728,729],[24,4212,584],{"className":4213},[104,105,729],[24,4215,737],{"className":4216},[736],[24,4218,4220],{"className":4219},[709],[24,4221,4223],{"className":4222,"style":744},[713],[24,4224],{},[24,4226,737],{"className":4227},[736],[24,4229,4231],{"className":4230},[709],[24,4232,4234],{"className":4233,"style":950},[713],[24,4235],{},[24,4237],{"className":4238},[152,891],[20,4240,4241,4242,238],{},"becomes more and more concentrated around ",[24,4243,4245,4258],{"className":4244,"translate":28},[32],[24,4246,4248],{"className":4247},[36],[38,4249,4250],{"xmlns":40},[43,4251,4252,4256],{},[46,4253,4254],{},[49,4255,322],{},[84,4257,322],{"encoding":86},[24,4259,4261],{"className":4260,"ariaHidden":72},[91],[24,4262,4264,4267],{"className":4263},[95],[24,4265],{"className":4266,"style":1374},[99],[24,4268,322],{"className":4269},[104,105],[20,4271,4272,4273,73],{},"As ",[24,4274,4276,4296],{"className":4275,"translate":28},[32],[24,4277,4279],{"className":4278},[36],[38,4280,4281],{"xmlns":40},[43,4282,4283,4293],{},[46,4284,4285,4287,4290],{},[49,4286,584],{},[53,4288,4289],{},"→",[49,4291,4292],{"mathvariant":368},"∞",[84,4294,4295],{"encoding":86},"n\\to\\infty",[24,4297,4299,4317],{"className":4298,"ariaHidden":72},[91],[24,4300,4302,4305,4308,4311,4314],{"className":4301},[95],[24,4303],{"className":4304,"style":1374},[99],[24,4306,584],{"className":4307},[104,105],[24,4309],{"className":4310,"style":111},[110],[24,4312,4289],{"className":4313},[115],[24,4315],{"className":4316,"style":111},[110],[24,4318,4320,4323],{"className":4319},[95],[24,4321],{"className":4322,"style":1374},[99],[24,4324,4292],{"className":4325},[104],[24,4327,4329],{"className":4328,"translate":28},[27],[24,4330,4332,4360],{"className":4331,"translate":28},[32],[24,4333,4335],{"className":4334},[36],[38,4336,4337],{"xmlns":40,"display":41},[43,4338,4339,4357],{},[46,4340,4341,4351,4353,4355],{},[628,4342,4343,4349],{},[577,4344,4345,4347],{},[49,4346,2594],{},[49,4348,584],{},[49,4350,584],{},[53,4352,4289],{},[49,4354,322],{},[49,4356,238],{"mathvariant":368},[84,4358,4359],{"encoding":86},"\\frac{X_n}{n}\n\\to x.",[24,4361,4363,4477],{"className":4362,"ariaHidden":72},[91],[24,4364,4366,4369,4468,4471,4474],{"className":4365},[95],[24,4367],{"className":4368,"style":4139},[99],[24,4370,4372,4375,4465],{"className":4371},[104],[24,4373],{"className":4374},[133,891],[24,4376,4378],{"className":4377},[628],[24,4379,4381,4457],{"className":4380},[704,705],[24,4382,4384,4454],{"className":4383},[709],[24,4385,4387,4398,4406],{"className":4386,"style":3269},[713],[24,4388,4389,4392],{"style":907},[24,4390],{"className":4391,"style":911},[721],[24,4393,4395],{"className":4394},[104],[24,4396,584],{"className":4397},[104,105],[24,4399,4400,4403],{"style":920},[24,4401],{"className":4402,"style":911},[721],[24,4404],{"className":4405,"style":928},[927],[24,4407,4408,4411],{"style":931},[24,4409],{"className":4410,"style":911},[721],[24,4412,4414],{"className":4413},[104],[24,4415,4417,4420],{"className":4416},[104],[24,4418,2594],{"className":4419,"style":2654},[104,105],[24,4421,4423],{"className":4422},[700],[24,4424,4426,4446],{"className":4425},[704,705],[24,4427,4429,4443],{"className":4428},[709],[24,4430,4432],{"className":4431,"style":714},[713],[24,4433,4434,4437],{"style":2669},[24,4435],{"className":4436,"style":722},[721],[24,4438,4440],{"className":4439},[726,727,728,729],[24,4441,584],{"className":4442},[104,105,729],[24,4444,737],{"className":4445},[736],[24,4447,4449],{"className":4448},[709],[24,4450,4452],{"className":4451,"style":744},[713],[24,4453],{},[24,4455,737],{"className":4456},[736],[24,4458,4460],{"className":4459},[709],[24,4461,4463],{"className":4462,"style":950},[713],[24,4464],{},[24,4466],{"className":4467},[152,891],[24,4469],{"className":4470,"style":111},[110],[24,4472,4289],{"className":4473},[115],[24,4475],{"className":4476,"style":111},[110],[24,4478,4480,4483,4486],{"className":4479},[95],[24,4481],{"className":4482,"style":1374},[99],[24,4484,322],{"className":4485},[104,105],[24,4487,238],{"className":4488},[104],[20,4490,4491],{},"Hence",[24,4493,4495],{"className":4494,"translate":28},[27],[24,4496,4498,4543],{"className":4497,"translate":28},[32],[24,4499,4501],{"className":4500},[36],[38,4502,4503],{"xmlns":40,"display":41},[43,4504,4505,4540],{},[46,4506,4507,4509,4511,4527,4530,4532,4534,4536,4538],{},[49,4508,51],{},[620,4510,622],{},[46,4512,4513,4515,4525],{},[53,4514,62],{"fence":72},[628,4516,4517,4523],{},[577,4518,4519,4521],{},[49,4520,2594],{},[49,4522,584],{},[49,4524,584],{},[53,4526,82],{"fence":72},[53,4528,4529],{},"≈",[49,4531,51],{},[53,4533,62],{"stretchy":61},[49,4535,322],{},[53,4537,82],{"stretchy":61},[49,4539,238],{"mathvariant":368},[84,4541,4542],{"encoding":86},"f\\!\\left(\\frac{X_n}{n}\\right)\n\\approx\nf(x).",[24,4544,4546,4684],{"className":4545,"ariaHidden":72},[91],[24,4547,4549,4552,4555,4558,4561,4675,4678,4681],{"className":4548},[95],[24,4550],{"className":4551,"style":1194},[99],[24,4553,51],{"className":4554,"style":106},[104,105],[24,4556],{"className":4557,"style":868},[110],[24,4559],{"className":4560,"style":145},[110],[24,4562,4564,4570,4669],{"className":4563},[875],[24,4565,4567],{"className":4566,"style":880},[133,879],[24,4568,62],{"className":4569},[884,728],[24,4571,4573,4576,4666],{"className":4572},[104],[24,4574],{"className":4575},[133,891],[24,4577,4579],{"className":4578},[628],[24,4580,4582,4658],{"className":4581},[704,705],[24,4583,4585,4655],{"className":4584},[709],[24,4586,4588,4599,4607],{"className":4587,"style":3269},[713],[24,4589,4590,4593],{"style":907},[24,4591],{"className":4592,"style":911},[721],[24,4594,4596],{"className":4595},[104],[24,4597,584],{"className":4598},[104,105],[24,4600,4601,4604],{"style":920},[24,4602],{"className":4603,"style":911},[721],[24,4605],{"className":4606,"style":928},[927],[24,4608,4609,4612],{"style":931},[24,4610],{"className":4611,"style":911},[721],[24,4613,4615],{"className":4614},[104],[24,4616,4618,4621],{"className":4617},[104],[24,4619,2594],{"className":4620,"style":2654},[104,105],[24,4622,4624],{"className":4623},[700],[24,4625,4627,4647],{"className":4626},[704,705],[24,4628,4630,4644],{"className":4629},[709],[24,4631,4633],{"className":4632,"style":714},[713],[24,4634,4635,4638],{"style":2669},[24,4636],{"className":4637,"style":722},[721],[24,4639,4641],{"className":4640},[726,727,728,729],[24,4642,584],{"className":4643},[104,105,729],[24,4645,737],{"className":4646},[736],[24,4648,4650],{"className":4649},[709],[24,4651,4653],{"className":4652,"style":744},[713],[24,4654],{},[24,4656,737],{"className":4657},[736],[24,4659,4661],{"className":4660},[709],[24,4662,4664],{"className":4663,"style":950},[713],[24,4665],{},[24,4667],{"className":4668},[152,891],[24,4670,4672],{"className":4671,"style":880},[152,879],[24,4673,82],{"className":4674},[884,728],[24,4676],{"className":4677,"style":111},[110],[24,4679,4529],{"className":4680},[115],[24,4682],{"className":4683,"style":111},[110],[24,4685,4687,4690,4693,4696,4699,4702],{"className":4686},[95],[24,4688],{"className":4689,"style":125},[99],[24,4691,51],{"className":4692,"style":106},[104,105],[24,4694,62],{"className":4695},[133],[24,4697,322],{"className":4698},[104,105],[24,4700,82],{"className":4701},[152],[24,4703,238],{"className":4704},[104],[20,4706,1938],{},[24,4708,4710],{"className":4709,"translate":28},[27],[24,4711,4713,4787],{"className":4712,"translate":28},[32],[24,4714,4716],{"className":4715},[36],[38,4717,4718],{"xmlns":40,"display":41},[43,4719,4720,4784],{},[46,4721,4722,4728,4730,4732,4734,4736,4738,4740,4742,4744,4746,4772,4774,4776,4778,4780,4782],{},[577,4723,4724,4726],{},[49,4725,581],{},[49,4727,584],{},[53,4729,62],{"stretchy":61},[49,4731,51],{},[53,4733,82],{"stretchy":61},[53,4735,62],{"stretchy":61},[49,4737,322],{},[53,4739,82],{"stretchy":61},[53,4741,599],{},[49,4743,3098],{},[620,4745,622],{},[46,4747,4748,4750,4752,4754,4770],{},[53,4749,65],{"fence":72},[49,4751,51],{},[620,4753,622],{},[46,4755,4756,4758,4768],{},[53,4757,62],{"fence":72},[628,4759,4760,4766],{},[577,4761,4762,4764],{},[49,4763,2594],{},[49,4765,584],{},[49,4767,584],{},[53,4769,82],{"fence":72},[53,4771,79],{"fence":72},[53,4773,4529],{},[49,4775,51],{},[53,4777,62],{"stretchy":61},[49,4779,322],{},[53,4781,82],{"stretchy":61},[49,4783,238],{"mathvariant":368},[84,4785,4786],{"encoding":86},"B_n(f)(x)\n=\nE\\!\\left[\nf\\!\\left(\\frac{X_n}{n}\\right)\n\\right]\n\\approx\nf(x).",[24,4788,4790,4863,5025],{"className":4789,"ariaHidden":72},[91],[24,4791,4793,4796,4836,4839,4842,4845,4848,4851,4854,4857,4860],{"className":4792},[95],[24,4794],{"className":4795,"style":125},[99],[24,4797,4799,4802],{"className":4798},[104],[24,4800,581],{"className":4801,"style":696},[104,105],[24,4803,4805],{"className":4804},[700],[24,4806,4808,4828],{"className":4807},[704,705],[24,4809,4811,4825],{"className":4810},[709],[24,4812,4814],{"className":4813,"style":714},[713],[24,4815,4816,4819],{"style":717},[24,4817],{"className":4818,"style":722},[721],[24,4820,4822],{"className":4821},[726,727,728,729],[24,4823,584],{"className":4824},[104,105,729],[24,4826,737],{"className":4827},[736],[24,4829,4831],{"className":4830},[709],[24,4832,4834],{"className":4833,"style":744},[713],[24,4835],{},[24,4837,62],{"className":4838},[133],[24,4840,51],{"className":4841,"style":106},[104,105],[24,4843,82],{"className":4844},[152],[24,4846,62],{"className":4847},[133],[24,4849,322],{"className":4850},[104,105],[24,4852,82],{"className":4853},[152],[24,4855],{"className":4856,"style":111},[110],[24,4858,599],{"className":4859},[115],[24,4861],{"className":4862,"style":111},[110],[24,4864,4866,4869,4872,4875,4878,5016,5019,5022],{"className":4865},[95],[24,4867],{"className":4868,"style":1194},[99],[24,4870,3098],{"className":4871,"style":3217},[104,105],[24,4873],{"className":4874,"style":868},[110],[24,4876],{"className":4877,"style":145},[110],[24,4879,4881,4887,4890,4893,4896,5010],{"className":4880},[875],[24,4882,4884],{"className":4883,"style":880},[133,879],[24,4885,65],{"className":4886},[884,728],[24,4888,51],{"className":4889,"style":106},[104,105],[24,4891],{"className":4892,"style":868},[110],[24,4894],{"className":4895,"style":145},[110],[24,4897,4899,4905,5004],{"className":4898},[875],[24,4900,4902],{"className":4901,"style":880},[133,879],[24,4903,62],{"className":4904},[884,728],[24,4906,4908,4911,5001],{"className":4907},[104],[24,4909],{"className":4910},[133,891],[24,4912,4914],{"className":4913},[628],[24,4915,4917,4993],{"className":4916},[704,705],[24,4918,4920,4990],{"className":4919},[709],[24,4921,4923,4934,4942],{"className":4922,"style":3269},[713],[24,4924,4925,4928],{"style":907},[24,4926],{"className":4927,"style":911},[721],[24,4929,4931],{"className":4930},[104],[24,4932,584],{"className":4933},[104,105],[24,4935,4936,4939],{"style":920},[24,4937],{"className":4938,"style":911},[721],[24,4940],{"className":4941,"style":928},[927],[24,4943,4944,4947],{"style":931},[24,4945],{"className":4946,"style":911},[721],[24,4948,4950],{"className":4949},[104],[24,4951,4953,4956],{"className":4952},[104],[24,4954,2594],{"className":4955,"style":2654},[104,105],[24,4957,4959],{"className":4958},[700],[24,4960,4962,4982],{"className":4961},[704,705],[24,4963,4965,4979],{"className":4964},[709],[24,4966,4968],{"className":4967,"style":714},[713],[24,4969,4970,4973],{"style":2669},[24,4971],{"className":4972,"style":722},[721],[24,4974,4976],{"className":4975},[726,727,728,729],[24,4977,584],{"className":4978},[104,105,729],[24,4980,737],{"className":4981},[736],[24,4983,4985],{"className":4984},[709],[24,4986,4988],{"className":4987,"style":744},[713],[24,4989],{},[24,4991,737],{"className":4992},[736],[24,4994,4996],{"className":4995},[709],[24,4997,4999],{"className":4998,"style":950},[713],[24,5000],{},[24,5002],{"className":5003},[152,891],[24,5005,5007],{"className":5006,"style":880},[152,879],[24,5008,82],{"className":5009},[884,728],[24,5011,5013],{"className":5012,"style":880},[152,879],[24,5014,79],{"className":5015},[884,728],[24,5017],{"className":5018,"style":111},[110],[24,5020,4529],{"className":5021},[115],[24,5023],{"className":5024,"style":111},[110],[24,5026,5028,5031,5034,5037,5040,5043],{"className":5027},[95],[24,5029],{"className":5030,"style":125},[99],[24,5032,51],{"className":5033,"style":106},[104,105],[24,5035,62],{"className":5036},[133],[24,5038,322],{"className":5039},[104,105],[24,5041,82],{"className":5042},[152],[24,5044,238],{"className":5045},[104],[20,5047,5048],{},"Intuitively, this holds.",[15,5050,5052],{"id":5051},"rigorous-proof","Rigorous Proof",[20,5054,5055,5056,5084,5085,5113,5114,5164,5165,5219],{},"Since ",[24,5057,5059,5072],{"className":5058,"translate":28},[32],[24,5060,5062],{"className":5061},[36],[38,5063,5064],{"xmlns":40},[43,5065,5066,5070],{},[46,5067,5068],{},[49,5069,51],{},[84,5071,51],{"encoding":86},[24,5073,5075],{"className":5074,"ariaHidden":72},[91],[24,5076,5078,5081],{"className":5077},[95],[24,5079],{"className":5080,"style":100},[99],[24,5082,51],{"className":5083,"style":106},[104,105]," is continuous on a closed interval, by properties of continuous functions on closed intervals: ",[24,5086,5088,5101],{"className":5087,"translate":28},[32],[24,5089,5091],{"className":5090},[36],[38,5092,5093],{"xmlns":40},[43,5094,5095,5099],{},[46,5096,5097],{},[49,5098,51],{},[84,5100,51],{"encoding":86},[24,5102,5104],{"className":5103,"ariaHidden":72},[91],[24,5105,5107,5110],{"className":5106},[95],[24,5108],{"className":5109,"style":100},[99],[24,5111,51],{"className":5112,"style":106},[104,105]," must be uniformly continuous. Therefore, for any ",[24,5115,5117,5134],{"className":5116,"translate":28},[32],[24,5118,5120],{"className":5119},[36],[38,5121,5122],{"xmlns":40},[43,5123,5124,5132],{},[46,5125,5126,5128,5130],{},[49,5127,259],{},[53,5129,262],{},[67,5131,69],{},[84,5133,267],{"encoding":86},[24,5135,5137,5155],{"className":5136,"ariaHidden":72},[91],[24,5138,5140,5143,5146,5149,5152],{"className":5139},[95],[24,5141],{"className":5142,"style":277},[99],[24,5144,259],{"className":5145},[104,105],[24,5147],{"className":5148,"style":111},[110],[24,5150,262],{"className":5151},[115],[24,5153],{"className":5154,"style":111},[110],[24,5156,5158,5161],{"className":5157},[95],[24,5159],{"className":5160,"style":296},[99],[24,5162,69],{"className":5163},[104]," there exists 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contribution does not exceed 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Then",[24,7074,7076],{"className":7075,"translate":28},[27],[24,7077,7079,7129],{"className":7078,"translate":28},[32],[24,7080,7082],{"className":7081},[36],[38,7083,7084],{"xmlns":40,"display":41},[43,7085,7086,7126],{},[46,7087,7088,7090,7092,7094,7100,7102,7104,7106,7108,7110,7112,7114,7116,7118,7120,7122,7124],{},[49,7089,369],{"mathvariant":368},[49,7091,51],{},[53,7093,62],{"stretchy":61},[577,7095,7096,7098],{},[49,7097,2594],{},[49,7099,584],{},[49,7101,5572],{"mathvariant":368},[49,7103,584],{},[53,7105,82],{"stretchy":61},[53,7107,380],{},[49,7109,51],{},[53,7111,62],{"stretchy":61},[49,7113,322],{},[53,7115,82],{"stretchy":61},[49,7117,369],{"mathvariant":368},[53,7119,5919],{},[67,7121,5358],{},[49,7123,6939],{},[49,7125,238],{"mathvariant":368},[84,7127,7128],{"encoding":86},"|f(X_n\u002Fn)-f(x)|\n\\le\n2M.",[24,7130,7132,7205,7235],{"className":7131,"ariaHidden":72},[91],[24,7133,7135,7138,7141,7144,7147,7187,7190,7193,7196,7199,7202],{"className":7134},[95],[24,7136],{"className":7137,"style":125},[99],[24,7139,369],{"className":7140},[104],[24,7142,51],{"className":7143,"style":106},[104,105],[24,7145,62],{"className":7146},[133],[24,7148,7150,7153],{"className":7149},[104],[24,7151,2594],{"className":7152,"style":2654},[104,105],[24,7154,7156],{"className":7155},[700],[24,7157,7159,7179],{"className":7158},[704,705],[24,7160,7162,7176],{"className":7161},[709],[24,7163,7165],{"className":7164,"style":714},[713],[24,7166,7167,7170],{"style":2669},[24,7168],{"className":7169,"style":722},[721],[24,7171,7173],{"className":7172},[726,727,728,729],[24,7174,584],{"className":7175},[104,105,729],[24,7177,737],{"className":7178},[736],[24,7180,7182],{"className":7181},[709],[24,7183,7185],{"className":7184,"style":744},[713],[24,7186],{},[24,7188,5572],{"className":7189},[104],[24,7191,584],{"className":7192},[104,105],[24,7194,82],{"className":7195},[152],[24,7197],{"className":7198,"style":450},[110],[24,7200,380],{"className":7201},[454],[24,7203],{"className":7204,"style":450},[110],[24,7206,7208,7211,7214,7217,7220,7223,7226,7229,7232],{"className":7207},[95],[24,7209],{"className":7210,"style":125},[99],[24,7212,51],{"className":7213,"style":106},[104,105],[24,7215,62],{"className":7216},[133],[24,7218,322],{"className":7219},[104,105],[24,7221,82],{"className":7222},[152],[24,7224,369],{"className":7225},[104],[24,7227],{"className":7228,"style":111},[110],[24,7230,5919],{"className":7231},[115],[24,7233],{"className":7234,"style":111},[110],[24,7236,7238,7241,7244,7247],{"className":7237},[95],[24,7239],{"className":7240,"style":6148},[99],[24,7242,5358],{"className":7243},[104],[24,7245,6939],{"className":7246,"style":6985},[104,105],[24,7248,238],{"className":7249},[104],[20,7251,7252,7253,238],{},"Thus the contribution does not exceed ",[24,7254,7256,7303],{"className":7255,"translate":28},[32],[24,7257,7259],{"className":7258},[36],[38,7260,7261],{"xmlns":40},[43,7262,7263,7300],{},[46,7264,7265,7267,7269,7272,7274,7276,7278,7284,7286,7288,7290,7292,7294,7296,7298],{},[67,7266,5358],{},[49,7268,6939],{},[53,7270,7271],{"separator":72},"⋅",[49,7273,317],{},[53,7275,62],{"stretchy":61},[49,7277,369],{"mathvariant":368},[577,7279,7280,7282],{},[49,7281,2594],{},[49,7283,584],{},[49,7285,5572],{"mathvariant":368},[49,7287,584],{},[53,7289,380],{},[49,7291,322],{},[49,7293,369],{"mathvariant":368},[53,7295,6295],{},[49,7297,5179],{},[53,7299,82],{"stretchy":61},[84,7301,7302],{"encoding":86},"2M·P(|X_n\u002Fn-x|\\ge\\delta)",[24,7304,7306,7388,7409],{"className":7305,"ariaHidden":72},[91],[24,7307,7309,7312,7315,7318,7321,7324,7327,7330,7333,7373,7376,7379,7382,7385],{"className":7308},[95],[24,7310],{"className":7311,"style":125},[99],[24,7313,5358],{"className":7314},[104],[24,7316,6939],{"className":7317,"style":6985},[104,105],[24,7319,7271],{"className":7320},[141],[24,7322],{"className":7323,"style":145},[110],[24,7325,317],{"className":7326,"style":340},[104,105],[24,7328,62],{"className":7329},[133],[24,7331,369],{"className":7332},[104],[24,7334,7336,7339],{"className":7335},[104],[24,7337,2594],{"className":7338,"style":2654},[104,105],[24,7340,7342],{"className":7341},[700],[24,7343,7345,7365],{"className":7344},[704,705],[24,7346,7348,7362],{"className":7347},[709],[24,7349,7351],{"className":7350,"style":714},[713],[24,7352,7353,7356],{"style":2669},[24,7354],{"className":7355,"style":722},[721],[24,7357,7359],{"className":7358},[726,727,728,729],[24,7360,584],{"className":7361},[104,105,729],[24,7363,737],{"className":7364},[736],[24,7366,7368],{"className":7367},[709],[24,7369,7371],{"className":7370,"style":744},[713],[24,7372],{},[24,7374,5572],{"className":7375},[104],[24,7377,584],{"className":7378},[104,105],[24,7380],{"className":7381,"style":450},[110],[24,7383,380],{"className":7384},[454],[24,7386],{"className":7387,"style":450},[110],[24,7389,7391,7394,7397,7400,7403,7406],{"className":7390},[95],[24,7392],{"className":7393,"style":125},[99],[24,7395,322],{"className":7396},[104,105],[24,7398,369],{"className":7399},[104],[24,7401],{"className":7402,"style":111},[110],[24,7404,6295],{"className":7405},[115],[24,7407],{"className":7408,"style":111},[110],[24,7410,7412,7415,7418],{"className":7411},[95],[24,7413],{"className":7414,"style":125},[99],[24,7416,5179],{"className":7417,"style":5200},[104,105],[24,7419,82],{"className":7420},[152],[20,7422,7423],{},"Using Chebyshev's inequality:",[24,7425,7427],{"className":7426,"translate":28},[27],[24,7428,7430,7519],{"className":7429,"translate":28},[32],[24,7431,7433],{"className":7432},[36],[38,7434,7435],{"xmlns":40,"display":41},[43,7436,7437,7516],{},[46,7438,7439,7441,7443,7445,7451,7453,7455,7457,7459,7461,7463,7465,7467,7469,7495,7497,7514],{},[49,7440,317],{},[53,7442,62],{"stretchy":61},[49,7444,369],{"mathvariant":368},[577,7446,7447,7449],{},[49,7448,2594],{},[49,7450,584],{},[49,7452,5572],{"mathvariant":368},[49,7454,584],{},[53,7456,380],{},[49,7458,322],{},[49,7460,369],{"mathvariant":368},[53,7462,6295],{},[49,7464,5179],{},[53,7466,82],{"stretchy":61},[53,7468,5919],{},[628,7470,7471,7485],{},[46,7472,7473,7475,7477,7479,7481,7483],{},[49,7474,322],{},[53,7476,62],{"stretchy":61},[67,7478,76],{},[53,7480,380],{},[49,7482,322],{},[53,7484,82],{"stretchy":61},[46,7486,7487,7489],{},[49,7488,584],{},[650,7490,7491,7493],{},[49,7492,5179],{},[67,7494,5358],{},[53,7496,5919],{},[628,7498,7499,7501],{},[67,7500,76],{},[46,7502,7503,7506,7508],{},[67,7504,7505],{},"4",[49,7507,584],{},[650,7509,7510,7512],{},[49,7511,5179],{},[67,7513,5358],{},[49,7515,238],{"mathvariant":368},[84,7517,7518],{"encoding":86},"P(|X_n\u002Fn-x|\\ge\\delta)\n\\le\n\\frac{x(1-x)}\n{n\\delta^2}\n\\le\n\\frac1{4n\\delta^2}.",[24,7520,7522,7592,7613,7634,7763],{"className":7521,"ariaHidden":72},[91],[24,7523,7525,7528,7531,7534,7537,7577,7580,7583,7586,7589],{"className":7524},[95],[24,7526],{"className":7527,"style":125},[99],[24,7529,317],{"className":7530,"style":340},[104,105],[24,7532,62],{"className":7533},[133],[24,7535,369],{"className":7536},[104],[24,7538,7540,7543],{"className":7539},[104],[24,7541,2594],{"className":7542,"style":2654},[104,105],[24,7544,7546],{"className":7545},[700],[24,7547,7549,7569],{"className":7548},[704,705],[24,7550,7552,7566],{"className":7551},[709],[24,7553,7555],{"className":7554,"style":714},[713],[24,7556,7557,7560],{"style":2669},[24,7558],{"className":7559,"style":722},[721],[24,7561,7563],{"className":7562},[726,727,728,729],[24,7564,584],{"className":7565},[104,105,729],[24,7567,737],{"className":7568},[736],[24,7570,7572],{"className":7571},[709],[24,7573,7575],{"className":7574,"style":744},[713],[24,7576],{},[24,7578,5572],{"className":7579},[104],[24,7581,584],{"className":7582},[104,105],[24,7584],{"className":7585,"style":450},[110],[24,7587,380],{"className":7588},[454],[24,7590],{"className":7591,"style":450},[110],[24,7593,7595,7598,7601,7604,7607,7610],{"className":7594},[95],[24,7596],{"className":7597,"style":125},[99],[24,7599,322],{"className":7600},[104,105],[24,7602,369],{"className":7603},[104],[24,7605],{"className":7606,"style":111},[110],[24,7608,6295],{"className":7609},[115],[24,7611],{"className":7612,"style":111},[110],[24,7614,7616,7619,7622,7625,7628,7631],{"className":7615},[95],[24,7617],{"className":7618,"style":125},[99],[24,7620,5179],{"className":7621,"style":5200},[104,105],[24,7623,82],{"className":7624},[152],[24,7626],{"className":7627,"style":111},[110],[24,7629,5919],{"className":7630},[115],[24,7632],{"className":7633,"style":111},[110],[24,7635,7637,7640,7754,7757,7760],{"className":7636},[95],[24,7638],{"className":7639,"style":4012},[99],[24,7641,7643,7646,7751],{"className":7642},[104],[24,7644],{"className":7645},[133,891],[24,7647,7649],{"className":7648},[628],[24,7650,7652,7743],{"className":7651},[704,705],[24,7653,7655,7740],{"className":7654},[709],[24,7656,7658,7700,7708],{"className":7657,"style":4031},[713],[24,7659,7660,7663],{"style":907},[24,7661],{"className":7662,"style":911},[721],[24,7664,7666,7669],{"className":7665},[104],[24,7667,584],{"className":7668},[104,105],[24,7670,7672,7675],{"className":7671},[104],[24,7673,5179],{"className":7674,"style":5200},[104,105],[24,7676,7678],{"className":7677},[700],[24,7679,7681],{"className":7680},[704],[24,7682,7684],{"className":7683},[709],[24,7685,7688],{"className":7686,"style":7687},[713],"height:0.7401em;",[24,7689,7691,7694],{"style":7690},"top:-2.989em;margin-right:0.05em;",[24,7692],{"className":7693,"style":722},[721],[24,7695,7697],{"className":7696},[726,727,728,729],[24,7698,5358],{"className":7699},[104,729],[24,7701,7702,7705],{"style":920},[24,7703],{"className":7704,"style":911},[721],[24,7706],{"className":7707,"style":928},[927],[24,7709,7710,7713],{"style":931},[24,7711],{"className":7712,"style":911},[721],[24,7714,7716,7719,7722,7725,7728,7731,7734,7737],{"className":7715},[104],[24,7717,322],{"className":7718},[104,105],[24,7720,62],{"className":7721},[133],[24,7723,76],{"className":7724},[104],[24,7726],{"className":7727,"style":450},[110],[24,7729,380],{"className":7730},[454],[24,7732],{"className":7733,"style":450},[110],[24,7735,322],{"className":7736},[104,105],[24,7738,82],{"className":7739},[152],[24,7741,737],{"className":7742},[736],[24,7744,7746],{"className":7745},[709],[24,7747,7749],{"className":7748,"style":950},[713],[24,7750],{},[24,7752],{"className":7753},[15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,[24,8081,8083],{"className":8082},[628],[24,8084,8086,8157],{"className":8085},[704,705],[24,8087,8089,8154],{"className":8088},[709],[24,8090,8092,8135,8143],{"className":8091,"style":3269},[713],[24,8093,8094,8097],{"style":907},[24,8095],{"className":8096,"style":911},[721],[24,8098,8100,8103,8106],{"className":8099},[104],[24,8101,5358],{"className":8102},[104],[24,8104,584],{"className":8105},[104,105],[24,8107,8109,8112],{"className":8108},[104],[24,8110,5179],{"className":8111,"style":5200},[104,105],[24,8113,8115],{"className":8114},[700],[24,8116,8118],{"className":8117},[704],[24,8119,8121],{"className":8120},[709],[24,8122,8124],{"className":8123,"style":7687},[713],[24,8125,8126,8129],{"style":7690},[24,8127],{"className":8128,"style":722},[721],[24,8130,8132],{"className":8131},[726,727,728,729],[24,8133,5358],{"className":8134},[104,729],[24,8136,8137,8140],{"style":920},[24,8138],{"className":8139,"style":911},[721],[24,8141],{"className":8142,"style":928},[927],[24,8144,8145,8148],{"style":931},[24,8146],{"className":8147,"style":911},[721],[24,8149,8151],{"className":8150},[104],[24,8152,6939],{"className":8153,"style":6985},[104,105],[24,8155,737],{"className":8156},[736],[24,8158,8160],{"className":8159},[709],[24,8161,8163],{"className":8162,"style":950},[713],[24,8164],{},[24,8166],{"className":8167},[152,891],[24,8169,238],{"className":8170},[104],[20,8172,8173,8174,8202],{},"When ",[24,8175,8177,8190],{"className":8176,"translate":28},[32],[24,8178,8180],{"className":8179},[36],[38,8181,8182],{"xmlns":40},[43,8183,8184,8188],{},[46,8185,8186],{},[49,8187,584],{},[84,8189,584],{"encoding":86},[24,8191,8193],{"className":8192,"ariaHidden":72},[91],[24,8194,8196,8199],{"className":8195},[95],[24,8197],{"className":8198,"style":1374},[99],[24,8200,584],{"className":8201},[104,105]," is sufficiently large,",[24,8204,8206],{"className":8205,"translate":28},[27],[24,8207,8209,8247],{"className":8208,"translate":28},[32],[24,8210,8212],{"className":8211},[36],[38,8213,8214],{"xmlns":40,"display":41},[43,8215,8216,8244],{},[46,8217,8218,8234,8236,8242],{},[628,8219,8220,8222],{},[49,8221,6939],{},[46,8223,8224,8226,8228],{},[67,8225,5358],{},[49,8227,584],{},[650,8229,8230,8232],{},[49,8231,5179],{},[67,8233,5358],{},[53,8235,393],{},[628,8237,8238,8240],{},[49,8239,259],{},[67,8241,5358],{},[49,8243,238],{"mathvariant":368},[84,8245,8246],{"encoding":86},"\\frac{M}{2n\\delta^2}\n\u003C\n\\frac{\\varepsilon}{2}.",[24,8248,8250,8359],{"className":8249,"ariaHidden":72},[91],[24,8251,8253,8256,8350,8353,8356],{"className":8252},[95],[24,8254],{"className":8255,"style":4139},[99],[24,8257,8259,8262,8347],{"className":8258},[104],[24,8260],{"className":8261},[133,891],[24,8263,8265],{"className":8264},[628],[24,8266,8268,8339],{"className":8267},[704,705],[24,8269,8271,8336],{"className":8270},[709],[24,8272,8274,8317,8325],{"className":8273,"style":3269},[713],[24,8275,8276,8279],{"style":907},[24,8277],{"className":8278,"style":911},[721],[24,8280,8282,8285,8288],{"className":8281},[104],[24,8283,5358],{"className":8284},[104],[24,8286,584],{"className":8287},[104,105],[24,8289,8291,8294],{"className":8290},[104],[24,8292,5179],{"className":8293,"style":5200},[104,105],[24,8295,8297],{"className":8296},[700],[24,8298,8300],{"className":8299},[704],[24,8301,8303],{"className":8302},[709],[24,8304,8306],{"className":8305,"style":7687},[713],[24,8307,8308,8311],{"style":7690},[24,8309],{"className":8310,"style":722},[721],[24,8312,8314],{"className":8313},[726,727,728,729],[24,8315,5358],{"className":8316},[104,729],[24,8318,8319,8322],{"style":920},[24,8320],{"className":8321,"style":911},[721],[24,8323],{"className":8324,"style":928},[927],[24,8326,8327,8330],{"style":931},[24,8328],{"className":8329,"style":911},[721],[24,8331,8333],{"className":8332},[104],[24,8334,6939],{"className":8335,"style":6985},[104,105],[24,8337,737],{"className":8338},[736],[24,8340,8342],{"className":8341},[709],[24,8343,8345],{"className":8344,"style":950},[713],[24,8346],{},[24,8348],{"className":8349},[152,891],[24,8351],{"className":8352,"style":111},[110],[24,8354,393],{"className":8355},[115],[24,8357],{"className":8358,"style":111},[110],[24,8360,8362,8365,8427],{"className":8361},[95],[24,8363],{"className":8364,"style":5433},[99],[24,8366,8368,8371,8424],{"className":8367},[104],[24,8369],{"className":8370},[133,891],[24,8372,8374],{"className":8373},[628],[24,8375,8377,8416],{"className":8376},[704,705],[24,8378,8380,8413],{"className":8379},[709],[24,8381,8383,8394,8402],{"className":8382,"style":986},[713],[24,8384,8385,8388],{"style":907},[24,8386],{"className":8387,"style":911},[721],[24,8389,8391],{"className":8390},[104],[24,8392,5358],{"className":8393},[104],[24,8395,8396,8399],{"style":920},[24,8397],{"className":8398,"style":911},[721],[24,8400],{"className":8401,"style":928},[927],[24,8403,8404,8407],{"style":931},[24,8405],{"className":8406,"style":911},[721],[24,8408,8410],{"className":8409},[104],[24,8411,259],{"className":8412},[104,105],[24,8414,737],{"className":8415},[736],[24,8417,8419],{"className":8418},[709],[24,8420,8422],{"className":8421,"style":950},[713],[24,8423],{},[24,8425],{"className":8426},[152,891],[24,8428,238],{"className":8429},[104],[20,8431,1938],{},[24,8433,8435],{"className":8434,"translate":28},[27],[24,8436,8438,8488],{"className":8437,"translate":28},[32],[24,8439,8441],{"className":8440},[36],[38,8442,8443],{"xmlns":40,"display":41},[43,8444,8445,8485],{},[46,8446,8447,8449,8455,8457,8459,8461,8463,8465,8467,8469,8471,8473,8475,8477,8479,8481,8483],{},[49,8448,369],{"mathvariant":368},[577,8450,8451,8453],{},[49,8452,581],{},[49,8454,584],{},[53,8456,62],{"stretchy":61},[49,8458,51],{},[53,8460,82],{"stretchy":61},[53,8462,62],{"stretchy":61},[49,8464,322],{},[53,8466,82],{"stretchy":61},[53,8468,380],{},[49,8470,51],{},[53,8472,62],{"stretchy":61},[49,8474,322],{},[53,8476,82],{"stretchy":61},[49,8478,369],{"mathvariant":368},[53,8480,393],{},[49,8482,259],{},[49,8484,238],{"mathvariant":368},[84,8486,8487],{"encoding":86},"|B_n(f)(x)-f(x)|\n\u003C\n\\varepsilon.",[24,8489,8491,8567,8597],{"className":8490,"ariaHidden":72},[91],[24,8492,8494,8497,8500,8540,8543,8546,8549,8552,8555,8558,8561,8564],{"className":8493},[95],[24,8495],{"className":8496,"style":125},[99],[24,8498,369],{"className":8499},[104],[24,8501,8503,8506],{"className":8502},[104],[24,8504,581],{"className":8505,"style":696},[104,105],[24,8507,8509],{"className":8508},[700],[24,8510,8512,8532],{"className":8511},[704,705],[24,8513,8515,8529],{"className":8514},[709],[24,8516,8518],{"className":8517,"style":714},[713],[24,8519,8520,8523],{"style":717},[24,8521],{"className":8522,"style":722},[721],[24,8524,8526],{"className":8525},[726,727,728,729],[24,8527,584],{"className":8528},[104,105,729],[24,8530,737],{"className":8531},[736],[24,8533,8535],{"className":8534},[709],[24,8536,8538],{"className":8537,"style":744},[713],[24,8539],{},[24,8541,62],{"className":8542},[133],[24,8544,51],{"className":8545,"style":106},[104,105],[24,8547,82],{"className":8548},[152],[24,8550,62],{"className":8551},[133],[24,8553,322],{"className":8554},[104,105],[24,8556,82],{"className":8557},[152],[24,8559],{"className":8560,"style":450},[110],[24,8562,380],{"className":8563},[454],[24,8565],{"className":8566,"style":450},[110],[24,8568,8570,8573,8576,8579,8582,8585,8588,8591,8594],{"className":8569},[95],[24,8571],{"className":8572,"style":125},[99],[24,8574,51],{"className":8575,"style":106},[104,105],[24,8577,62],{"className":8578},[133],[24,8580,322],{"className":8581},[104,105],[24,8583,82],{"className":8584},[152],[24,8586,369],{"className":8587},[104],[24,8589],{"className":8590,"style":111},[110],[24,8592,393],{"className":8593},[115],[24,8595],{"className":8596,"style":111},[110],[24,8598,8600,8603,8606],{"className":8599},[95],[24,8601],{"className":8602,"style":1374},[99],[24,8604,259],{"className":8605},[104,105],[24,8607,238],{"className":8608},[104],[20,8610,8611,8612,238],{},"Moreover, this estimate is independent of ",[24,8613,8615,8628],{"className":8614,"translate":28},[32],[24,8616,8618],{"className":8617},[36],[38,8619,8620],{"xmlns":40},[43,8621,8622,8626],{},[46,8623,8624],{},[49,8625,322],{},[84,8627,322],{"encoding":86},[24,8629,8631],{"className":8630,"ariaHidden":72},[91],[24,8632,8634,8637],{"className":8633},[95],[24,8635],{"className":8636,"style":1374},[99],[24,8638,322],{"className":8639},[104,105],[20,8641,8642],{},"Consequently,",[24,8644,8646],{"className":8645,"translate":28},[27],[24,8647,8649,8724],{"className":8648,"translate":28},[32],[24,8650,8652],{"className":8651},[36],[38,8653,8654],{"xmlns":40,"display":41},[43,8655,8656,8721],{},[46,8657,8658,8684,8686,8692,8694,8696,8698,8700,8702,8704,8706,8708,8710,8712,8714,8716,8718],{},[8659,8660,8661,8668],"munder",{},[46,8662,8663,8666],{},[49,8664,8665],{},"sup",[53,8667,6951],{},[46,8669,8670,8672,8674,8676,8678,8680,8682],{},[49,8671,322],{},[53,8673,55],{},[53,8675,65],{"stretchy":61},[67,8677,69],{},[53,8679,73],{"separator":72},[67,8681,76],{},[53,8683,79],{"stretchy":61},[49,8685,369],{"mathvariant":368},[577,8687,8688,8690],{},[49,8689,581],{},[49,8691,584],{},[53,8693,62],{"stretchy":61},[49,8695,51],{},[53,8697,82],{"stretchy":61},[53,8699,62],{"stretchy":61},[49,8701,322],{},[53,8703,82],{"stretchy":61},[53,8705,380],{},[49,8707,51],{},[53,8709,62],{"stretchy":61},[49,8711,322],{},[53,8713,82],{"stretchy":61},[49,8715,369],{"mathvariant":368},[53,8717,4289],{},[67,8719,8720],{},"0.",[84,8722,8723],{"encoding":86},"\\sup_{x\\in[0,1]}\n|B_n(f)(x)-f(x)|\n\\to 0.",[24,8725,8727,8875,8905],{"className":8726,"ariaHidden":72},[91],[24,8728,8730,8734,8802,8805,8808,8848,8851,8854,8857,8860,8863,8866,8869,8872],{"className":8729},[95],[24,8731],{"className":8732,"style":8733},[99],"height:1.9104em;vertical-align:-1.1604em;",[24,8735,8737],{"className":8736},[784,785],[24,8738,8740,8793],{"className":8739},[704,705],[24,8741,8743,8790],{"className":8742},[709],[24,8744,8746,8779],{"className":8745,"style":1374},[713],[24,8747,8749,8752],{"style":8748},"top:-2.1146em;margin-left:0em;",[24,8750],{"className":8751,"style":911},[721],[24,8753,8755],{"className":8754},[726,727,728,729],[24,8756,8758,8761,8764,8767,8770,8773,8776],{"className":8757},[104,729],[24,8759,322],{"className":8760},[104,105,729],[24,8762,55],{"className":8763},[115,729],[24,8765,65],{"className":8766},[133,729],[24,8768,69],{"className":8769},[104,729],[24,8771,73],{"className":8772},[141,729],[24,8774,76],{"className":8775},[104,729],[24,8777,79],{"className":8778},[152,729],[24,8780,8782,8785],{"style":8781},"top:-3em;",[24,8783],{"className":8784,"style":911},[721],[24,8786,8787],{},[24,8788,8665],{"className":8789},[784],[24,8791,737],{"className":8792},[736],[24,8794,8796],{"className":8795},[709],[24,8797,8800],{"className":8798,"style":8799},[713],"height:1.1604em;",[24,8801],{},[24,8803],{"className":8804,"style":145},[110],[24,8806,369],{"className":8807},[104],[24,8809,8811,8814],{"className":8810},[104],[24,8812,581],{"className":8813,"style":696},[104,105],[24,8815,8817],{"className":8816},[700],[24,8818,8820,8840],{"className":8819},[704,705],[24,8821,8823,8837],{"className":8822},[709],[24,8824,8826],{"className":8825,"style":714},[713],[24,8827,8828,8831],{"style":717},[24,8829],{"className":8830,"style":722},[721],[24,8832,8834],{"className":8833},[726,727,728,729],[24,8835,584],{"className":8836},[104,105,729],[24,8838,737],{"className":8839},[736],[24,8841,8843],{"className":8842},[709],[24,8844,8846],{"className":8845,"style":744},[713],[24,8847],{},[24,8849,62],{"className":8850},[133],[24,8852,51],{"className":8853,"style":106},[104,105],[24,8855,82],{"className":8856},[152],[24,8858,62],{"className":8859},[133],[24,8861,322],{"className":8862},[104,105],[24,8864,82],{"className":8865},[152],[24,8867],{"className":8868,"style":450},[110],[24,8870,380],{"className":8871},[454],[24,8873],{"className":8874,"style":450},[110],[24,8876,8878,8881,8884,8887,8890,8893,8896,8899,8902],{"className":8877},[95],[24,8879],{"className":8880,"style":125},[99],[24,8882,51],{"className":8883,"style":106},[104,105],[24,8885,62],{"className":8886},[133],[24,8888,322],{"className":8889},[104,105],[24,8891,82],{"className":8892},[152],[24,8894,369],{"className":8895},[104],[24,8897],{"className":8898,"style":111},[110],[24,8900,4289],{"className":8901},[115],[24,8903],{"className":8904,"style":111},[110],[24,8906,8908,8911],{"className":8907},[95],[24,8909],{"className":8910,"style":296},[99],[24,8912,8720],{"className":8913},[104],[20,8915,2736],{},[24,8917,8919],{"className":8918,"translate":28},[27],[24,8920,8922,8950],{"className":8921,"translate":28},[32],[24,8923,8925],{"className":8924},[36],[38,8926,8927],{"xmlns":40,"display":41},[43,8928,8929,8947],{},[46,8930,8931,8937,8939,8941,8943,8945],{},[577,8932,8933,8935],{},[49,8934,581],{},[49,8936,584],{},[53,8938,62],{"stretchy":61},[49,8940,51],{},[53,8942,82],{"stretchy":61},[53,8944,4289],{},[49,8946,51],{},[84,8948,8949],{"encoding":86},"B_n(f)\n\\to f",[24,8951,8953,9017],{"className":8952,"ariaHidden":72},[91],[24,8954,8956,8959,8999,9002,9005,9008,9011,9014],{"className":8955},[95],[24,8957],{"className":8958,"style":125},[99],[24,8960,8962,8965],{"className":8961},[104],[24,8963,581],{"className":8964,"style":696},[104,105],[24,8966,8968],{"className":8967},[700],[24,8969,8971,8991],{"className":8970},[704,705],[24,8972,8974,8988],{"className":8973},[709],[24,8975,8977],{"className":8976,"style":714},[713],[24,8978,8979,8982],{"style":717},[24,8980],{"className":8981,"style":722},[721],[24,8983,8985],{"className":8984},[726,727,728,729],[24,8986,584],{"className":8987},[104,105,729],[24,8989,737],{"className":8990},[736],[24,8992,8994],{"className":8993},[709],[24,8995,8997],{"className":8996,"style":744},[713],[24,8998],{},[24,9000,62],{"className":9001},[133],[24,9003,51],{"className":9004,"style":106},[104,105],[24,9006,82],{"className":9007},[152],[24,9009],{"className":9010,"style":111},[110],[24,9012,4289],{"className":9013},[115],[24,9015],{"className":9016,"style":111},[110],[24,9018,9020,9023],{"className":9019},[95],[24,9021],{"className":9022,"style":100},[99],[24,9024,51],{"className":9025,"style":106},[104,105],[20,9027,9028],{},"uniformly.",[20,9030,9031],{},"Weierstrass theorem is proved.",[10,9033,9035],{"id":9034},"proof-of-the-uat","Proof of the UAT",[20,9037,9038],{},"Next, we will go through the process of proving the Universal Approximation Theorem (UAT) using polynomial approximation, step by step, in language understandable with basic calculus. We take the univariate continuous function as an example for clarity; the vector case can be generalized similarly.",[20,9040,9041,9044,9045,9147,9148,9200],{},[555,9042,9043],{},"Goal:"," Given a continuous function ",[24,9046,9048,9082],{"className":9047,"translate":28},[32],[24,9049,9051],{"className":9050},[36],[38,9052,9053],{"xmlns":40},[43,9054,9055,9079],{},[46,9056,9057,9059,9062,9064,9066,9068,9071,9073,9075],{},[49,9058,51],{},[53,9060,9061],{},":",[53,9063,65],{"stretchy":61},[49,9065,2619],{},[53,9067,73],{"separator":72},[49,9069,9070],{},"b",[53,9072,79],{"stretchy":61},[53,9074,4289],{},[49,9076,9078],{"mathvariant":9077},"double-struck","R",[84,9080,9081],{"encoding":86},"f:[a,b]\\to \\mathbb{R}",[24,9083,9085,9103,9136],{"className":9084,"ariaHidden":72},[91],[24,9086,9088,9091,9094,9097,9100],{"className":9087},[95],[24,9089],{"className":9090,"style":100},[99],[24,9092,51],{"className":9093,"style":106},[104,105],[24,9095],{"className":9096,"style":111},[110],[24,9098,9061],{"className":9099},[115],[24,9101],{"className":9102,"style":111},[110],[24,9104,9106,9109,9112,9115,9118,9121,9124,9127,9130,9133],{"className":9105},[95],[24,9107],{"className":9108,"style":125},[99],[24,9110,65],{"className":9111},[133],[24,9113,2619],{"className":9114},[104,105],[24,9116,73],{"className":9117},[141],[24,9119],{"className":9120,"style":145},[110],[24,9122,9070],{"className":9123},[104,105],[24,9125,79],{"className":9126},[152],[24,9128],{"className":9129,"style":111},[110],[24,9131,4289],{"className":9132},[115],[24,9134],{"className":9135,"style":111},[110],[24,9137,9139,9143],{"className":9138},[95],[24,9140],{"className":9141,"style":9142},[99],"height:0.6889em;",[24,9144,9078],{"className":9145},[104,9146],"mathbb"," and any ",[24,9149,9151,9170],{"className":9150,"translate":28},[32],[24,9152,9154],{"className":9153},[36],[38,9155,9156],{"xmlns":40},[43,9157,9158,9167],{},[46,9159,9160,9163,9165],{},[49,9161,9162],{},"ϵ",[53,9164,262],{},[67,9166,69],{},[84,9168,9169],{"encoding":86},"\\epsilon>0",[24,9171,9173,9191],{"className":9172,"ariaHidden":72},[91],[24,9174,9176,9179,9182,9185,9188],{"className":9175},[95],[24,9177],{"className":9178,"style":277},[99],[24,9180,9162],{"className":9181},[104,105],[24,9183],{"className":9184,"style":111},[110],[24,9186,262],{"className":9187},[115],[24,9189],{"className":9190,"style":111},[110],[24,9192,9194,9197],{"className":9193},[95],[24,9195],{"className":9196,"style":296},[99],[24,9198,69],{"className":9199},[104],", there exists a single-hidden-layer neural network",[24,9202,9204],{"className":9203,"translate":28},[27],[24,9205,9207,9276],{"className":9206,"translate":28},[32],[24,9208,9210],{"className":9209},[36],[38,9211,9212],{"xmlns":40,"display":41},[43,9213,9214,9273],{},[46,9215,9216,9219,9221,9223,9225,9227,9241,9248,9251,9253,9260,9262,9264,9271],{},[49,9217,9218],{},"N",[53,9220,62],{"stretchy":61},[49,9222,322],{},[53,9224,82],{"stretchy":61},[53,9226,599],{},[601,9228,9229,9231,9239],{},[53,9230,605],{},[46,9232,9233,9235,9237],{},[49,9234,2606],{},[53,9236,599],{},[67,9238,76],{},[49,9240,584],{},[577,9242,9243,9246],{},[49,9244,9245],{},"α",[49,9247,2606],{},[49,9249,9250],{},"σ",[53,9252,62],{"stretchy":61},[577,9254,9255,9258],{},[49,9256,9257],{},"β",[49,9259,2606],{},[49,9261,322],{},[53,9263,1571],{},[577,9265,9266,9269],{},[49,9267,9268],{},"γ",[49,9270,2606],{},[53,9272,82],{"stretchy":61},[84,9274,9275],{"encoding":86},"N(x) = \\sum_{i=1}^{n} \\alpha_i \\sigma(\\beta_i x + \\gamma_i)",[24,9277,9279,9306,9488],{"className":9278,"ariaHidden":72},[91],[24,9280,9282,9285,9288,9291,9294,9297,9300,9303],{"className":9281},[95],[24,9283],{"className":9284,"style":125},[99],[24,9286,9218],{"className":9287,"style":6985},[104,105],[24,9289,62],{"className":9290},[133],[24,9292,322],{"className":9293},[104,105],[24,9295,82],{"className":9296},[152],[24,9298],{"className":9299,"style":111},[110],[24,9301,599],{"className":9302},[115],[24,9304],{"className":9305,"style":111},[110],[24,9307,9309,9313,9382,9385,9428,9431,9434,9476,9479,9482,9485],{"className":9308},[95],[24,9310],{"className":9311,"style":9312},[99],"height:2.9291em;vertical-align:-1.2777em;",[24,9314,9316],{"className":9315},[784,785],[24,9317,9319,9373],{"className":9318},[704,705],[24,9320,9322,9370],{"className":9321},[709],[24,9323,9325,9346,9356],{"className":9324,"style":795},[713],[24,9326,9328,9331],{"style":9327},"top:-1.8723em;margin-left:0em;",[24,9329],{"className":9330,"style":802},[721],[24,9332,9334],{"className":9333},[726,727,728,729],[24,9335,9337,9340,9343],{"className":9336},[104,729],[24,9338,2606],{"className":9339},[104,105,729],[24,9341,599],{"className":9342},[115,729],[24,9344,76],{"className":9345},[104,729],[24,9347,9348,9351],{"style":821},[24,9349],{"className":9350,"style":802},[721],[24,9352,9353],{},[24,9354,605],{"className":9355},[784,830,831],[24,9357,9358,9361],{"style":834},[24,9359],{"className":9360,"style":802},[721],[24,9362,9364],{"className":9363},[726,727,728,729],[24,9365,9367],{"className":9366},[104,729],[24,9368,584],{"className":9369},[104,105,729],[24,9371,737],{"className":9372},[736],[24,9374,9376],{"className":9375},[709],[24,9377,9380],{"className":9378,"style":9379},[713],"height:1.2777em;",[24,9381],{},[24,9383],{"className":9384,"style":145},[110],[24,9386,9388,9392],{"className":9387},[104],[24,9389,9245],{"className":9390,"style":9391},[104,105],"margin-right:0.0037em;",[24,9393,9395],{"className":9394},[700],[24,9396,9398,9420],{"className":9397},[704,705],[24,9399,9401,9417],{"className":9400},[709],[24,9402,9405],{"className":9403,"style":9404},[713],"height:0.3117em;",[24,9406,9408,9411],{"style":9407},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[24,9409],{"className":9410,"style":722},[721],[24,9412,9414],{"className":9413},[726,727,728,729],[24,9415,2606],{"className":9416},[104,105,729],[24,9418,737],{"className":9419},[736],[24,9421,9423],{"className":9422},[709],[24,9424,9426],{"className":9425,"style":744},[713],[24,9427],{},[24,9429,9250],{"className":9430,"style":5287},[104,105],[24,9432,62],{"className":9433},[133],[24,9435,9437,9441],{"className":9436},[104],[24,9438,9257],{"className":9439,"style":9440},[104,105],"margin-right:0.0528em;",[24,9442,9444],{"className":9443},[700],[24,9445,9447,9468],{"className":9446},[704,705],[24,9448,9450,9465],{"className":9449},[709],[24,9451,9453],{"className":9452,"style":9404},[713],[24,9454,9456,9459],{"style":9455},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[24,9457],{"className":9458,"style":722},[721],[24,9460,9462],{"className":9461},[726,727,728,729],[24,9463,2606],{"className":9464},[104,105,729],[24,9466,737],{"className":9467},[736],[24,9469,9471],{"className":9470},[709],[24,9472,9474],{"className":9473,"style":744},[713],[24,9475],{},[24,9477,322],{"className":9478},[104,105],[24,9480],{"className":9481,"style":450},[110],[24,9483,1571],{"className":9484},[454],[24,9486],{"className":9487,"style":450},[110],[24,9489,9491,9494,9536],{"className":9490},[95],[24,9492],{"className":9493,"style":125},[99],[24,9495,9497,9501],{"className":9496},[104],[24,9498,9268],{"className":9499,"style":9500},[104,105],"margin-right:0.0556em;",[24,9502,9504],{"className":9503},[700],[24,9505,9507,9528],{"className":9506},[704,705],[24,9508,9510,9525],{"className":9509},[709],[24,9511,9513],{"className":9512,"style":9404},[713],[24,9514,9516,9519],{"style":9515},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[24,9517],{"className":9518,"style":722},[721],[24,9520,9522],{"className":9521},[726,727,728,729],[24,9523,2606],{"className":9524},[104,105,729],[24,9526,737],{"className":9527},[736],[24,9529,9531],{"className":9530},[709],[24,9532,9534],{"className":9533,"style":744},[713],[24,9535],{},[24,9537,82],{"className":9538},[152],[20,9540,9541],{},"such that",[24,9543,9545],{"className":9544,"translate":28},[27],[24,9546,9548,9607],{"className":9547,"translate":28},[32],[24,9549,9551],{"className":9550},[36],[38,9552,9553],{"xmlns":40,"display":41},[43,9554,9555,9604],{},[46,9556,9557,9559,9561,9563,9565,9567,9569,9571,9573,9575,9577,9579,9581,9583,9585,9588,9590,9592,9594,9596,9598,9600,9602],{},[49,9558,369],{"mathvariant":368},[49,9560,9218],{},[53,9562,62],{"stretchy":61},[49,9564,322],{},[53,9566,82],{"stretchy":61},[53,9568,380],{},[49,9570,51],{},[53,9572,62],{"stretchy":61},[49,9574,322],{},[53,9576,82],{"stretchy":61},[49,9578,369],{"mathvariant":368},[53,9580,393],{},[49,9582,9162],{},[53,9584,73],{"separator":72},[110,9586],{"width":9587},"1em",[49,9589,403],{"mathvariant":368},[49,9591,322],{},[53,9593,55],{},[53,9595,65],{"stretchy":61},[49,9597,2619],{},[53,9599,73],{"separator":72},[49,9601,9070],{},[53,9603,79],{"stretchy":61},[84,9605,9606],{"encoding":86},"|N(x) - f(x)| \u003C \\epsilon, \\quad \\forall x\\in [a,b]",[24,9608,9610,9640,9670,9704],{"className":9609,"ariaHidden":72},[91],[24,9611,9613,9616,9619,9622,9625,9628,9631,9634,9637],{"className":9612},[95],[24,9614],{"className":9615,"style":125},[99],[24,9617,369],{"className":9618},[104],[24,9620,9218],{"className":9621,"style":6985},[104,105],[24,9623,62],{"className":9624},[133],[24,9626,322],{"className":9627},[104,105],[24,9629,82],{"className":9630},[152],[24,9632],{"className":9633,"style":450},[110],[24,9635,380],{"className":9636},[454],[24,9638],{"className":9639,"style":450},[110],[24,9641,9643,9646,9649,9652,9655,9658,9661,9664,9667],{"className":9642},[95],[24,9644],{"className":9645,"style":125},[99],[24,9647,51],{"className":9648,"style":106},[104,105],[24,9650,62],{"className":9651},[133],[24,9653,322],{"className":9654},[104,105],[24,9656,82],{"className":9657},[152],[24,9659,369],{"className":9660},[104],[24,9662],{"className":9663,"style":111},[110],[24,9665,393],{"className":9666},[115],[24,9668],{"className":9669,"style":111},[110],[24,9671,9673,9676,9679,9682,9686,9689,9692,9695,9698,9701],{"className":9672},[95],[24,9674],{"className":9675,"style":100},[99],[24,9677,9162],{"className":9678},[104,105],[24,9680,73],{"className":9681},[141],[24,9683],{"className":9684,"style":9685},[110],"margin-right:1em;",[24,9687],{"className":9688,"style":145},[110],[24,9690,403],{"className":9691},[104],[24,9693,322],{"className":9694},[104,105],[24,9696],{"className":9697,"style":111},[110],[24,9699,55],{"className":9700},[115],[24,9702],{"className":9703,"style":111},[110],[24,9705,9707,9710,9713,9716,9719,9722,9725],{"className":9706},[95],[24,9708],{"className":9709,"style":125},[99],[24,9711,65],{"className":9712},[133],[24,9714,2619],{"className":9715},[104,105],[24,9717,73],{"className":9718},[141],[24,9720],{"className":9721,"style":145},[110],[24,9723,9070],{"className":9724},[104,105],[24,9726,79],{"className":9727},[152],[20,9729,9730,9731,9760],{},"where ",[24,9732,9734,9748],{"className":9733,"translate":28},[32],[24,9735,9737],{"className":9736},[36],[38,9738,9739],{"xmlns":40},[43,9740,9741,9745],{},[46,9742,9743],{},[49,9744,9250],{},[84,9746,9747],{"encoding":86},"\\sigma",[24,9749,9751],{"className":9750,"ariaHidden":72},[91],[24,9752,9754,9757],{"className":9753},[95],[24,9755],{"className":9756,"style":1374},[99],[24,9758,9250],{"className":9759,"style":5287},[104,105]," is a \"non-polynomial, continuous\" activation function (e.g., sigmoid).",[20,9762,9763],{},[555,9764,9765],{},"Step 1: Use the Weierstrass Theorem",[20,9767,9768,9769,9812],{},"We have already proved this theorem earlier: any continuous function can be approximated by a polynomial ",[24,9770,9772,9791],{"className":9771,"translate":28},[32],[24,9773,9775],{"className":9774},[36],[38,9776,9777],{"xmlns":40},[43,9778,9779,9789],{},[46,9780,9781,9783,9785,9787],{},[49,9782,317],{},[53,9784,62],{"stretchy":61},[49,9786,322],{},[53,9788,82],{"stretchy":61},[84,9790,327],{"encoding":86},[24,9792,9794],{"className":9793,"ariaHidden":72},[91],[24,9795,9797,9800,9803,9806,9809],{"className":9796},[95],[24,9798],{"className":9799,"style":125},[99],[24,9801,317],{"className":9802,"style":340},[104,105],[24,9804,62],{"className":9805},[133],[24,9807,322],{"className":9808},[104,105],[24,9810,82],{"className":9811},[152],". 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As we know, a polynomial can be written as ",[24,10017,10019,10066],{"className":10018,"translate":28},[32],[24,10020,10022],{"className":10021},[36],[38,10023,10024],{"xmlns":40},[43,10025,10026,10063],{},[46,10027,10028,10034,10036,10042,10044,10046,10049,10051,10057],{},[577,10029,10030,10032],{},[49,10031,2619],{},[67,10033,69],{},[53,10035,1571],{},[577,10037,10038,10040],{},[49,10039,2619],{},[67,10041,76],{},[49,10043,322],{},[53,10045,1571],{},[53,10047,10048],{},"⋯",[53,10050,1571],{},[577,10052,10053,10055],{},[49,10054,2619],{},[49,10056,2614],{},[650,10058,10059,10061],{},[49,10060,322],{},[49,10062,2614],{},[84,10064,10065],{"encoding":86},"a_0 + a_1 x + \\dots + a_m x^m",[24,10067,10069,10126,10184,10203],{"className":10068,"ariaHidden":72},[91],[24,10070,10072,10076,10117,10120,10123],{"className":10071},[95],[24,10073],{"className":10074,"style":10075},[99],"height:0.7333em;vertical-align:-0.15em;",[24,10077,10079,10082],{"className":10078},[104],[24,10080,2619],{"className":10081},[104,105],[24,10083,10085],{"className":10084},[700],[24,10086,10088,10109],{"className":10087},[704,705],[24,10089,10091,10106],{"className":10090},[709],[24,10092,10095],{"className":10093,"style":10094},[713],"height:0.3011em;",[24,10096,10097,10100],{"style":2357},[24,10098],{"className":10099,"style":722},[721],[24,10101,10103],{"className":10102},[726,727,728,729],[24,10104,69],{"className":10105},[104,729],[24,10107,737],{"className":10108},[736],[24,10110,10112],{"className":10111},[709],[24,10113,10115],{"className":10114,"style":744},[713],[24,10116],{},[24,10118],{"className":10119,"style":450},[110],[24,10121,1571],{"className":10122},[454],[24,10124],{"className":10125,"style":450},[110],[24,10127,10129,10132,10172,10175,10178,10181],{"className":10128},[95],[24,10130],{"className":10131,"style":10075},[99],[24,10133,10135,10138],{"className":10134},[104],[24,10136,2619],{"className":10137},[104,105],[24,10139,10141],{"className":10140},[700],[24,10142,10144,10164],{"className":10143},[704,705],[24,10145,10147,10161],{"className":10146},[709],[24,10148,10150],{"className":10149,"style":10094},[713],[24,10151,10152,10155],{"style":2357},[24,10153],{"className":10154,"style":722},[721],[24,10156,10158],{"className":10157},[726,727,728,729],[24,10159,76],{"className":10160},[104,729],[24,10162,737],{"className":10163},[736],[24,10165,10167],{"className":10166},[709],[24,10168,10170],{"className":10169,"style":744},[713],[24,10171],{},[24,10173,322],{"className":10174},[104,105],[24,10176],{"className":10177,"style":450},[110],[24,10179,1571],{"className":10180},[454],[24,10182],{"className":10183,"style":450},[110],[24,10185,10187,10191,10194,10197,10200],{"className":10186},[95],[24,10188],{"className":10189,"style":10190},[99],"height:0.6667em;vertical-align:-0.0833em;",[24,10192,10048],{"className":10193},[875],[24,10195],{"className":10196,"style":450},[110],[24,10198,1571],{"className":10199},[454],[24,10201],{"className":10202,"style":450},[110],[24,10204,10206,10210,10250],{"className":10205},[95],[24,10207],{"className":10208,"style":10209},[99],"height:0.8144em;vertical-align:-0.15em;",[24,10211,10213,10216],{"className":10212},[104],[24,10214,2619],{"className":10215},[104,105],[24,10217,10219],{"className":10218},[700],[24,10220,10222,10242],{"className":10221},[704,705],[24,10223,10225,10239],{"className":10224},[709],[24,10226,10228],{"className":10227,"style":714},[713],[24,10229,10230,10233],{"style":2357},[24,10231],{"className":10232,"style":722},[721],[24,10234,10236],{"className":10235},[726,727,728,729],[24,10237,2614],{"className":10238},[104,105,729],[24,10240,737],{"className":10241},[736],[24,10243,10245],{"className":10244},[709],[24,10246,10248],{"className":10247,"style":744},[713],[24,10249],{},[24,10251,10253,10256],{"className":10252},[104],[24,10254,322],{"className":10255},[104,105],[24,10257,10259],{"className":10258},[700],[24,10260,10262],{"className":10261},[704],[24,10263,10265],{"className":10264},[709],[24,10266,10269],{"className":10267,"style":10268},[713],"height:0.6644em;",[24,10270,10272,10275],{"style":10271},"top:-3.063em;margin-right:0.05em;",[24,10273],{"className":10274,"style":722},[721],[24,10276,10278],{"className":10277},[726,727,728,729],[24,10279,2614],{"className":10280},[104,105,729],[20,10282,10283],{},[555,10284,10285,10286,10346],{},"Step 2: Approximate monomials ",[24,10287,10289,10307],{"className":10288,"translate":28},[32],[24,10290,10292],{"className":10291},[36],[38,10293,10294],{"xmlns":40},[43,10295,10296,10304],{},[46,10297,10298],{},[650,10299,10300,10302],{},[49,10301,322],{},[49,10303,610],{},[84,10305,10306],{"encoding":86},"x^k",[24,10308,10310],{"className":10309,"ariaHidden":72},[91],[24,10311,10313,10317],{"className":10312},[95],[24,10314],{"className":10315,"style":10316},[99],"height:0.8491em;",[24,10318,10320,10323],{"className":10319},[104],[24,10321,322],{"className":10322},[104,105],[24,10324,10326],{"className":10325},[700],[24,10327,10329],{"className":10328},[704],[24,10330,10332],{"className":10331},[709],[24,10333,10335],{"className":10334,"style":10316},[713],[24,10336,10337,10340],{"style":10271},[24,10338],{"className":10339,"style":722},[721],[24,10341,10343],{"className":10342},[726,727,728,729],[24,10344,610],{"className":10345,"style":812},[104,105,729]," using an activation function",[20,10348,10349,10350,10408,10409,238],{},"If we can approximate ",[24,10351,10353,10370],{"className":10352,"translate":28},[32],[24,10354,10356],{"className":10355},[36],[38,10357,10358],{"xmlns":40},[43,10359,10360,10368],{},[46,10361,10362],{},[650,10363,10364,10366],{},[49,10365,322],{},[49,10367,610],{},[84,10369,10306],{"encoding":86},[24,10371,10373],{"className":10372,"ariaHidden":72},[91],[24,10374,10376,10379],{"className":10375},[95],[24,10377],{"className":10378,"style":10316},[99],[24,10380,10382,10385],{"className":10381},[104],[24,10383,322],{"className":10384},[104,105],[24,10386,10388],{"className":10387},[700],[24,10389,10391],{"className":10390},[704],[24,10392,10394],{"className":10393},[709],[24,10395,10397],{"className":10396,"style":10316},[713],[24,10398,10399,10402],{"style":10271},[24,10400],{"className":10401,"style":722},[721],[24,10403,10405],{"className":10404},[726,727,728,729],[24,10406,610],{"className":10407,"style":812},[104,105,729]," by a linear combination of neurons in one hidden layer, then we can approximate the polynomial ",[24,10410,10412,10431],{"className":10411,"translate":28},[32],[24,10413,10415],{"className":10414},[36],[38,10416,10417],{"xmlns":40},[43,10418,10419,10429],{},[46,10420,10421,10423,10425,10427],{},[49,10422,317],{},[53,10424,62],{"stretchy":61},[49,10426,322],{},[53,10428,82],{"stretchy":61},[84,10430,327],{"encoding":86},[24,10432,10434],{"className":10433,"ariaHidden":72},[91],[24,10435,10437,10440,10443,10446,10449],{"className":10436},[95],[24,10438],{"className":10439,"style":125},[99],[24,10441,317],{"className":10442,"style":340},[104,105],[24,10444,62],{"className":10445},[133],[24,10447,322],{"className":10448},[104,105],[24,10450,82],{"className":10451},[152],[20,10453,10454],{},"The single-hidden-layer network has the form:",[24,10456,10458],{"className":10457,"translate":28},[27],[24,10459,10461,10525],{"className":10460,"translate":28},[32],[24,10462,10464],{"className":10463},[36],[38,10465,10466],{"xmlns":40,"display":41},[43,10467,10468,10522],{},[46,10469,10470,10472,10474,10476,10478,10480,10494,10500,10502,10504,10510,10512,10514,10520],{},[49,10471,9218],{},[53,10473,62],{"stretchy":61},[49,10475,322],{},[53,10477,82],{"stretchy":61},[53,10479,599],{},[601,10481,10482,10484,10492],{},[53,10483,605],{},[46,10485,10486,10488,10490],{},[49,10487,2606],{},[53,10489,599],{},[67,10491,76],{},[49,10493,584],{},[577,10495,10496,10498],{},[49,10497,9245],{},[49,10499,2606],{},[49,10501,9250],{},[53,10503,62],{"stretchy":61},[577,10505,10506,10508],{},[49,10507,9257],{},[49,10509,2606],{},[49,10511,322],{},[53,10513,1571],{},[577,10515,10516,10518],{},[49,10517,9268],{},[49,10519,2606],{},[53,10521,82],{"stretchy":61},[84,10523,10524],{"encoding":86},"  N(x) = \\sum_{i=1}^{n} \\alpha_i \\sigma(\\beta_i x + \\gamma_i)",[24,10526,10528,10555,10729],{"className":10527,"ariaHidden":72},[91],[24,10529,10531,10534,10537,10540,10543,10546,10549,10552],{"className":10530},[95],[24,10532],{"className":10533,"style":125},[99],[24,10535,9218],{"className":10536,"style":6985},[104,105],[24,10538,62],{"className":10539},[133],[24,10541,322],{"className":10542},[104,105],[24,10544,82],{"className":10545},[152],[24,10547],{"className":10548,"style":111},[110],[24,10550,599],{"className":10551},[115],[24,10553],{"className":10554,"style":111},[110],[24,10556,10558,10561,10628,10631,10671,10674,10677,10717,10720,10723,10726],{"className":10557},[95],[24,10559],{"className":10560,"style":9312},[99],[24,10562,10564],{"className":10563},[784,785],[24,10565,10567,10620],{"className":10566},[704,705],[24,10568,10570,10617],{"className":10569},[709],[24,10571,10573,10593,10603],{"className":10572,"style":795},[713],[24,10574,10575,10578],{"style":9327},[24,10576],{"className":10577,"style":802},[721],[24,10579,10581],{"className":10580},[726,727,728,729],[24,10582,10584,10587,10590],{"className":10583},[104,729],[24,10585,2606],{"className":10586},[104,105,729],[24,10588,599],{"className":10589},[115,729],[24,10591,76],{"className":10592},[104,729],[24,10594,10595,10598],{"style":821},[24,10596],{"className":10597,"style":802},[721],[24,10599,10600],{},[24,10601,605],{"className":10602},[784,830,831],[24,10604,10605,10608],{"style":834},[24,10606],{"className":10607,"style":802},[721],[24,10609,10611],{"className":10610},[726,727,728,729],[24,10612,10614],{"className":10613},[104,729],[24,10615,584],{"className":10616},[104,105,729],[24,10618,737],{"className":10619},[736],[24,10621,10623],{"className":10622},[709],[24,10624,10626],{"className":10625,"style":9379},[713],[24,10627],{},[24,10629],{"className":10630,"style":145},[110],[24,10632,10634,10637],{"className":10633},[104],[24,10635,9245],{"className":10636,"style":9391},[104,105],[24,10638,10640],{"className":10639},[700],[24,10641,10643,10663],{"className":10642},[704,705],[24,10644,10646,10660],{"className":10645},[709],[24,10647,10649],{"className":10648,"style":9404},[713],[24,10650,10651,10654],{"style":9407},[24,10652],{"className":10653,"style":722},[721],[24,10655,10657],{"className":10656},[726,727,728,729],[24,10658,2606],{"className":10659},[104,105,729],[24,10661,737],{"className":10662},[736],[24,10664,10666],{"className":10665},[709],[24,10667,10669],{"className":10668,"style":744},[713],[24,10670],{},[24,10672,9250],{"className":10673,"style":5287},[104,105],[24,10675,62],{"className":10676},[133],[24,10678,10680,10683],{"className":10679},[104],[24,10681,9257],{"className":10682,"style":9440},[104,105],[24,10684,10686],{"className":10685},[700],[24,10687,10689,10709],{"className":10688},[704,705],[24,10690,10692,10706],{"className":10691},[709],[24,10693,10695],{"className":10694,"style":9404},[713],[24,10696,10697,10700],{"style":9455},[24,10698],{"className":10699,"style":722},[721],[24,10701,10703],{"className":10702},[726,727,728,729],[24,10704,2606],{"className":10705},[104,105,729],[24,10707,737],{"className":10708},[736],[24,10710,10712],{"className":10711},[709],[24,10713,10715],{"className":10714,"style":744},[713],[24,10716],{},[24,10718,322],{"className":10719},[104,105],[24,10721],{"className":10722,"style":450},[110],[24,10724,1571],{"className":10725},[454],[24,10727],{"className":10728,"style":450},[110],[24,10730,10732,10735,10775],{"className":10731},[95],[24,10733],{"className":10734,"style":125},[99],[24,10736,10738,10741],{"className":10737},[104],[24,10739,9268],{"className":10740,"style":9500},[104,105],[24,10742,10744],{"className":10743},[700],[24,10745,10747,10767],{"className":10746},[704,705],[24,10748,10750,10764],{"className":10749},[709],[24,10751,10753],{"className":10752,"style":9404},[713],[24,10754,10755,10758],{"style":9515},[24,10756],{"className":10757,"style":722},[721],[24,10759,10761],{"className":10760},[726,727,728,729],[24,10762,2606],{"className":10763},[104,105,729],[24,10765,737],{"className":10766},[736],[24,10768,10770],{"className":10769},[709],[24,10771,10773],{"className":10772,"style":744},[713],[24,10774],{},[24,10776,82],{"className":10777},[152],[20,10779,10780,10781,10825],{},"Choose a suitable ",[24,10782,10784,10804],{"className":10783,"translate":28},[32],[24,10785,10787],{"className":10786},[36],[38,10788,10789],{"xmlns":40},[43,10790,10791,10801],{},[46,10792,10793,10795,10797,10799],{},[49,10794,9250],{},[53,10796,62],{"stretchy":61},[49,10798,322],{},[53,10800,82],{"stretchy":61},[84,10802,10803],{"encoding":86},"\\sigma(x)",[24,10805,10807],{"className":10806,"ariaHidden":72},[91],[24,10808,10810,10813,10816,10819,10822],{"className":10809},[95],[24,10811],{"className":10812,"style":125},[99],[24,10814,9250],{"className":10815,"style":5287},[104,105],[24,10817,62],{"className":10818},[133],[24,10820,322],{"className":10821},[104,105],[24,10823,82],{"className":10824},[152],", such as the sigmoid:",[24,10827,10829],{"className":10828,"translate":28},[27],[24,10830,10832,10875],{"className":10831,"translate":28},[32],[24,10833,10835],{"className":10834},[36],[38,10836,10837],{"xmlns":40,"display":41},[43,10838,10839,10872],{},[46,10840,10841,10843,10845,10847,10849,10851],{},[49,10842,9250],{},[53,10844,62],{"stretchy":61},[49,10846,322],{},[53,10848,82],{"stretchy":61},[53,10850,599],{},[628,10852,10853,10855],{},[67,10854,76],{},[46,10856,10857,10859,10861],{},[67,10858,76],{},[53,10860,1571],{},[650,10862,10863,10866],{},[49,10864,10865],{},"e",[46,10867,10868,10870],{},[53,10869,380],{},[49,10871,322],{},[84,10873,10874],{"encoding":86},"   \\sigma(x) = \\frac{1}{1+e^{-x}}",[24,10876,10878,10905],{"className":10877,"ariaHidden":72},[91],[24,10879,10881,10884,10887,10890,10893,10896,10899,10902],{"className":10880},[95],[24,10882],{"className":10883,"style":125},[99],[24,10885,9250],{"className":10886,"style":5287},[104,105],[24,10888,62],{"className":10889},[133],[24,10891,322],{"className":10892},[104,105],[24,10894,82],{"className":10895},[152],[24,10897],{"className":10898,"style":111},[110],[24,10900,599],{"className":10901},[115],[24,10903],{"className":10904,"style":111},[110],[24,10906,10908,10912],{"className":10907},[95],[24,10909],{"className":10910,"style":10911},[99],"height:2.0908em;vertical-align:-0.7693em;",[24,10913,10915,10918,11017],{"className":10914},[104],[24,10916],{"className":10917},[133,891],[24,10919,10921],{"className":10920},[628],[24,10922,10924,11008],{"className":10923},[704,705],[24,10925,10927,11005],{"className":10926},[709],[24,10928,10930,10986,10994],{"className":10929,"style":2161},[713],[24,10931,10932,10935],{"style":907},[24,10933],{"className":10934,"style":911},[721],[24,10936,10938,10941,10944,10947,10950],{"className":10937},[104],[24,10939,76],{"className":10940},[104],[24,10942],{"className":10943,"style":450},[110],[24,10945,1571],{"className":10946},[454],[24,10948],{"className":10949,"style":450},[110],[24,10951,10953,10956],{"className":10952},[104],[24,10954,10865],{"className":10955},[104,105],[24,10957,10959],{"className":10958},[700],[24,10960,10962],{"className":10961},[704],[24,10963,10965],{"className":10964},[709],[24,10966,10969],{"className":10967,"style":10968},[713],"height:0.6973em;",[24,10970,10971,10974],{"style":7690},[24,10972],{"className":10973,"style":722},[721],[24,10975,10977],{"className":10976},[726,727,728,729],[24,10978,10980,10983],{"className":10979},[104,729],[24,10981,380],{"className":10982},[104,729],[24,10984,322],{"className":10985},[104,105,729],[24,10987,10988,10991],{"style":920},[24,10989],{"className":10990,"style":911},[721],[24,10992],{"className":10993,"style":928},[927],[24,10995,10996,10999],{"style":931},[24,10997],{"className":10998,"style":911},[721],[24,11000,11002],{"className":11001},[104],[24,11003,76],{"className":11004},[104],[24,11006,737],{"className":11007},[736],[24,11009,11011],{"className":11010},[709],[24,11012,11015],{"className":11013,"style":11014},[713],"height:0.7693em;",[24,11016],{},[24,11018],{"className":11019},[152,891],[20,11021,11022],{},"Its Taylor expansion is non-zero locally:",[24,11024,11026],{"className":11025,"translate":28},[27],[24,11027,11029,11087],{"className":11028,"translate":28},[32],[24,11030,11032],{"className":11031},[36],[38,11033,11034],{"xmlns":40,"display":41},[43,11035,11036,11084],{},[46,11037,11038,11040,11042,11044,11046,11048,11054,11056,11062,11064,11066,11073,11080,11082],{},[49,11039,9250],{},[53,11041,62],{"stretchy":61},[49,11043,322],{},[53,11045,82],{"stretchy":61},[53,11047,599],{},[628,11049,11050,11052],{},[67,11051,76],{},[67,11053,5358],{},[53,11055,1571],{},[628,11057,11058,11060],{},[67,11059,76],{},[67,11061,7505],{},[49,11063,322],{},[53,11065,380],{},[628,11067,11068,11070],{},[67,11069,76],{},[67,11071,11072],{},"48",[650,11074,11075,11077],{},[49,11076,322],{},[67,11078,11079],{},"3",[53,11081,1571],{},[53,11083,2087],{},[84,11085,11086],{"encoding":86},"  \\sigma(x) = \\frac12 + \\frac14 x - \\frac1{48} x^3 + \\dots",[24,11088,11090,11117,11194,11274,11381],{"className":11089,"ariaHidden":72},[91],[24,11091,11093,11096,11099,11102,11105,11108,11111,11114],{"className":11092},[95],[24,11094],{"className":11095,"style":125},[99],[24,11097,9250],{"className":11098,"style":5287},[104,105],[24,11100,62],{"className":11101},[133],[24,11103,322],{"className":11104},[104,105],[24,11106,82],{"className":11107},[152],[24,11109],{"className":11110,"style":111},[110],[24,11112,599],{"className":11113},[115],[24,11115],{"className":11116,"style":111},[110],[24,11118,11120,11123,11185,11188,11191],{"className":11119},[95],[24,11121],{"className":11122,"style":7769},[99],[24,11124,11126,11129,11182],{"className":11125},[104],[24,11127],{"className":11128},[133,891],[24,11130,11132],{"className":11131},[628],[24,11133,11135,11174],{"className":11134},[704,705],[24,11136,11138,11171],{"className":11137},[709],[24,11139,11141,11152,11160],{"className":11140,"style":2161},[713],[24,11142,11143,11146],{"style":907},[24,11144],{"className":11145,"style":911},[721],[24,11147,11149],{"className":11148},[104],[24,11150,5358],{"className":11151},[104],[24,11153,11154,11157],{"style":920},[24,11155],{"className":11156,"style":911},[721],[24,11158],{"className":11159,"style":928},[927],[24,11161,11162,11165],{"style":931},[24,11163],{"className":11164,"style":911},[721],[24,11166,11168],{"className":11167},[104],[24,11169,76],{"className":11170},[104],[24,11172,737],{"className":11173},[736],[24,11175,11177],{"className":11176},[709],[24,11178,11180],{"className":11179,"style":950},[713],[24,11181],{},[24,11183],{"className":11184},[152,891],[24,11186],{"className":11187,"style":450},[110],[24,11189,1571],{"className":11190},[454],[24,11192],{"className":11193,"style":450},[110],[24,11195,11197,11200,11262,11265,11268,11271],{"className":11196},[95],[24,11198],{"className":11199,"style":7769},[99],[24,11201,11203,11206,11259],{"className":11202},[104],[24,11204],{"className":11205},[133,891],[24,11207,11209],{"className":11208},[628],[24,11210,11212,11251],{"className":11211},[704,705],[24,11213,11215,11248],{"className":11214},[709],[24,11216,11218,11229,11237],{"className":11217,"style":2161},[713],[24,11219,11220,11223],{"style":907},[24,11221],{"className":11222,"style":911},[721],[24,11224,11226],{"className":11225},[104],[24,11227,7505],{"className":11228},[104],[24,11230,11231,11234],{"style":920},[24,11232],{"className":11233,"style":911},[721],[24,11235],{"className":11236,"style":928},[927],[24,11238,11239,11242],{"style":931},[24,11240],{"className":11241,"style":911},[721],[24,11243,11245],{"className":11244},[104],[24,11246,76],{"className":11247},[104],[24,11249,737],{"className":11250},[736],[24,11252,11254],{"className":11253},[709],[24,11255,11257],{"className":11256,"style":950},[713],[24,11258],{},[24,11260],{"className":11261},[152,891],[24,11263,322],{"className":11264},[104,105],[24,11266],{"className":11267,"style":450},[110],[24,11269,380],{"className":11270},[454],[24,11272],{"className":11273,"style":450},[110],[24,11275,11277,11280,11342,11372,11375,11378],{"className":11276},[95],[24,11278],{"className":11279,"style":7769},[99],[24,11281,11283,11286,11339],{"className":11282},[104],[24,11284],{"className":11285},[133,891],[24,11287,11289],{"className":11288},[628],[24,11290,11292,11331],{"className":11291},[704,705],[24,11293,11295,11328],{"className":11294},[709],[24,11296,11298,11309,11317],{"className":11297,"style":2161},[713],[24,11299,11300,11303],{"style":907},[24,11301],{"className":11302,"style":911},[721],[24,11304,11306],{"className":11305},[104],[24,11307,11072],{"className":11308},[104],[24,11310,11311,11314],{"style":920},[24,11312],{"className":11313,"style":911},[721],[24,11315],{"className":11316,"style":928},[927],[24,11318,11319,11322],{"style":931},[24,11320],{"className":11321,"style":911},[721],[24,11323,11325],{"className":11324},[104],[24,11326,76],{"className":11327},[104],[24,11329,737],{"className":11330},[736],[24,11332,11334],{"className":11333},[709],[24,11335,11337],{"className":11336,"style":950},[713],[24,11338],{},[24,11340],{"className":11341},[152,891],[24,11343,11345,11348],{"className":11344},[104],[24,11346,322],{"className":11347},[104,105],[24,11349,11351],{"className":11350},[700],[24,11352,11354],{"className":11353},[704],[24,11355,11357],{"className":11356},[709],[24,11358,11361],{"className":11359,"style":11360},[713],"height:0.8641em;",[24,11362,11363,11366],{"style":1047},[24,11364],{"className":11365,"style":722},[721],[24,11367,11369],{"className":11368},[726,727,728,729],[24,11370,11079],{"className":11371},[104,729],[24,11373],{"className":11374,"style":450},[110],[24,11376,1571],{"className":11377},[454],[24,11379],{"className":11380,"style":450},[110],[24,11382,11384,11388],{"className":11383},[95],[24,11385],{"className":11386,"style":11387},[99],"height:0.123em;",[24,11389,2087],{"className":11390},[875],[20,11392,11393],{},"One can generate arbitrary polynomial terms using linear combinations of its shifted versions.",[20,11395,11396,11397,11575],{},"By adjusting ",[24,11398,11400,11434],{"className":11399,"translate":28},[32],[24,11401,11403],{"className":11402},[36],[38,11404,11405],{"xmlns":40},[43,11406,11407,11431],{},[46,11408,11409,11415,11417,11423,11425],{},[577,11410,11411,11413],{},[49,11412,9245],{},[49,11414,2606],{},[53,11416,73],{"separator":72},[577,11418,11419,11421],{},[49,11420,9257],{},[49,11422,2606],{},[53,11424,73],{"separator":72},[577,11426,11427,11429],{},[49,11428,9268],{},[49,11430,2606],{},[84,11432,11433],{"encoding":86},"\\alpha_i, \\beta_i, \\gamma_i",[24,11435,11437],{"className":11436,"ariaHidden":72},[91],[24,11438,11440,11443,11483,11486,11489,11529,11532,11535],{"className":11439},[95],[24,11441],{"className":11442,"style":100},[99],[24,11444,11446,11449],{"className":11445},[104],[24,11447,9245],{"className":11448,"style":9391},[104,105],[24,11450,11452],{"className":11451},[700],[24,11453,11455,11475],{"className":11454},[704,705],[24,11456,11458,11472],{"className":11457},[709],[24,11459,11461],{"className":11460,"style":9404},[713],[24,11462,11463,11466],{"style":9407},[24,11464],{"className":11465,"style":722},[721],[24,11467,11469],{"className":11468},[726,727,728,729],[24,11470,2606],{"className":11471},[104,105,729],[24,11473,737],{"className":11474},[736],[24,11476,11478],{"className":11477},[709],[24,11479,11481],{"className":11480,"style":744},[713],[24,11482],{},[24,11484,73],{"className":11485},[141],[24,11487],{"className":11488,"style":145},[110],[24,11490,11492,11495],{"className":11491},[104],[24,11493,9257],{"className":11494,"style":9440},[104,105],[24,11496,11498],{"className":11497},[700],[24,11499,11501,11521],{"className":11500},[704,705],[24,11502,11504,11518],{"className":11503},[709],[24,11505,11507],{"className":11506,"style":9404},[713],[24,11508,11509,11512],{"style":9455},[24,11510],{"className":11511,"style":722},[721],[24,11513,11515],{"className":11514},[726,727,728,729],[24,11516,2606],{"className":11517},[104,105,729],[24,11519,737],{"className":11520},[736],[24,11522,11524],{"className":11523},[709],[24,11525,11527],{"className":11526,"style":744},[713],[24,11528],{},[24,11530,73],{"className":11531},[141],[24,11533],{"className":11534,"style":145},[110],[24,11536,11538,11541],{"className":11537},[104],[24,11539,9268],{"className":11540,"style":9500},[104,105],[24,11542,11544],{"className":11543},[700],[24,11545,11547,11567],{"className":11546},[704,705],[24,11548,11550,11564],{"className":11549},[709],[24,11551,11553],{"className":11552,"style":9404},[713],[24,11554,11555,11558],{"style":9515},[24,11556],{"className":11557,"style":722},[721],[24,11559,11561],{"className":11560},[726,727,728,729],[24,11562,2606],{"className":11563},[104,105,729],[24,11565,737],{"className":11566},[736],[24,11568,11570],{"className":11569},[709],[24,11571,11573],{"className":11572,"style":744},[713],[24,11574],{},", the linear combination",[24,11577,11579],{"className":11578,"translate":28},[27],[24,11580,11582,11628],{"className":11581,"translate":28},[32],[24,11583,11585],{"className":11584},[36],[38,11586,11587],{"xmlns":40,"display":41},[43,11588,11589,11625],{},[46,11590,11591,11597,11603,11605,11607,11613,11615,11617,11623],{},[8659,11592,11593,11595],{},[53,11594,605],{},[49,11596,2606],{},[577,11598,11599,11601],{},[49,11600,9245],{},[49,11602,2606],{},[49,11604,9250],{},[53,11606,62],{"stretchy":61},[577,11608,11609,11611],{},[49,11610,9257],{},[49,11612,2606],{},[49,11614,322],{},[53,11616,1571],{},[577,11618,11619,11621],{},[49,11620,9268],{},[49,11622,2606],{},[53,11624,82],{"stretchy":61},[84,11626,11627],{"encoding":86},"   \\sum_i \\alpha_i \\sigma(\\beta_i x + \\gamma_i)",[24,11629,11631,11784],{"className":11630,"ariaHidden":72},[91],[24,11632,11634,11638,11683,11686,11726,11729,11732,11772,11775,11778,11781],{"className":11633},[95],[24,11635],{"className":11636,"style":11637},[99],"height:2.3277em;vertical-align:-1.2777em;",[24,11639,11641],{"className":11640},[784,785],[24,11642,11644,11675],{"className":11643},[704,705],[24,11645,11647,11672],{"className":11646},[709],[24,11648,11651,11662],{"className":11649,"style":11650},[713],"height:1.05em;",[24,11652,11653,11656],{"style":9327},[24,11654],{"className":11655,"style":802},[721],[24,11657,11659],{"className":11658},[726,727,728,729],[24,11660,2606],{"className":11661},[104,105,729],[24,11663,11664,11667],{"style":821},[24,11665],{"className":11666,"style":802},[721],[24,11668,11669],{},[24,11670,605],{"className":11671},[784,830,831],[24,11673,737],{"className":11674},[736],[24,11676,11678],{"className":11677},[709],[24,11679,11681],{"className":11680,"style":9379},[713],[24,11682],{},[24,11684],{"className":11685,"style":145},[110],[24,11687,11689,11692],{"className":11688},[104],[24,11690,9245],{"className":11691,"style":9391},[104,105],[24,11693,11695],{"className":11694},[700],[24,11696,11698,11718],{"className":11697},[704,705],[24,11699,11701,11715],{"className":11700},[709],[24,11702,11704],{"className":11703,"style":9404},[713],[24,11705,11706,11709],{"style":9407},[24,11707],{"className":11708,"style":722},[721],[24,11710,11712],{"className":11711},[726,727,728,729],[24,11713,2606],{"className":11714},[104,105,729],[24,11716,737],{"className":11717},[736],[24,11719,11721],{"className":11720},[709],[24,11722,11724],{"className":11723,"style":744},[713],[24,11725],{},[24,11727,9250],{"className":11728,"style":5287},[104,105],[24,11730,62],{"className":11731},[133],[24,11733,11735,11738],{"className":11734},[104],[24,11736,9257],{"className":11737,"style":9440},[104,105],[24,11739,11741],{"className":11740},[700],[24,11742,11744,11764],{"className":11743},[704,705],[24,11745,11747,11761],{"className":11746},[709],[24,11748,11750],{"className":11749,"style":9404},[713],[24,11751,11752,11755],{"style":9455},[24,11753],{"className":11754,"style":722},[721],[24,11756,11758],{"className":11757},[726,727,728,729],[24,11759,2606],{"className":11760},[104,105,729],[24,11762,737],{"className":11763},[736],[24,11765,11767],{"className":11766},[709],[24,11768,11770],{"className":11769,"style":744},[713],[24,11771],{},[24,11773,322],{"className":11774},[104,105],[24,11776],{"className":11777,"style":450},[110],[24,11779,1571],{"className":11780},[454],[24,11782],{"className":11783,"style":450},[110],[24,11785,11787,11790,11830],{"className":11786},[95],[24,11788],{"className":11789,"style":125},[99],[24,11791,11793,11796],{"className":11792},[104],[24,11794,9268],{"className":11795,"style":9500},[104,105],[24,11797,11799],{"className":11798},[700],[24,11800,11802,11822],{"className":11801},[704,705],[24,11803,11805,11819],{"className":11804},[709],[24,11806,11808],{"className":11807,"style":9404},[713],[24,11809,11810,11813],{"style":9515},[24,11811],{"className":11812,"style":722},[721],[24,11814,11816],{"className":11815},[726,727,728,729],[24,11817,2606],{"className":11818},[104,105,729],[24,11820,737],{"className":11821},[736],[24,11823,11825],{"className":11824},[709],[24,11826,11828],{"className":11827,"style":744},[713],[24,11829],{},[24,11831,82],{"className":11832},[152],[20,11834,11835,11836,11894],{},"can approximate ",[24,11837,11839,11856],{"className":11838,"translate":28},[32],[24,11840,11842],{"className":11841},[36],[38,11843,11844],{"xmlns":40},[43,11845,11846,11854],{},[46,11847,11848],{},[650,11849,11850,11852],{},[49,11851,322],{},[49,11853,610],{},[84,11855,10306],{"encoding":86},[24,11857,11859],{"className":11858,"ariaHidden":72},[91],[24,11860,11862,11865],{"className":11861},[95],[24,11863],{"className":11864,"style":10316},[99],[24,11866,11868,11871],{"className":11867},[104],[24,11869,322],{"className":11870},[104,105],[24,11872,11874],{"className":11873},[700],[24,11875,11877],{"className":11876},[704],[24,11878,11880],{"className":11879},[709],[24,11881,11883],{"className":11882,"style":10316},[713],[24,11884,11885,11888],{"style":10271},[24,11886],{"className":11887,"style":722},[721],[24,11889,11891],{"className":11890},[726,727,728,729],[24,11892,610],{"className":11893,"style":812},[104,105,729]," to arbitrary precision (similar to Taylor polynomial approximation in calculus).",[11896,11897,11898],"blockquote",{},[20,11899,11900,11903],{},[555,11901,11902],{},"Important intuition here:"," Each neuron is a \"nonlinear function block\", and summing many such blocks can combine into a polynomial form.",[20,11905,11906],{},[555,11907,11908],{},"Step 3: Combine to approximate the polynomial",[20,11910,11911,11912,12139,12140,12198,12199,12285],{},"For a polynomial ",[24,11913,11915,11964],{"className":11914,"translate":28},[32],[24,11916,11918],{"className":11917},[36],[38,11919,11920],{"xmlns":40},[43,11921,11922,11961],{},[46,11923,11924,11926,11928,11930,11932,11934,11949,11955],{},[49,11925,317],{},[53,11927,62],{"stretchy":61},[49,11929,322],{},[53,11931,82],{"stretchy":61},[53,11933,599],{},[11935,11936,11937,11939,11947],"msubsup",{},[53,11938,605],{},[46,11940,11941,11943,11945],{},[49,11942,610],{},[53,11944,599],{},[67,11946,69],{},[49,11948,2614],{},[577,11950,11951,11953],{},[49,11952,2619],{},[49,11954,610],{},[650,11956,11957,11959],{},[49,11958,322],{},[49,11960,610],{},[84,11962,11963],{"encoding":86},"P(x) = \\sum_{k=0}^m a_k x^k",[24,11965,11967,11994],{"className":11966,"ariaHidden":72},[91],[24,11968,11970,11973,11976,11979,11982,11985,11988,11991],{"className":11969},[95],[24,11971],{"className":11972,"style":125},[99],[24,11974,317],{"className":11975,"style":340},[104,105],[24,11977,62],{"className":11978},[133],[24,11980,322],{"className":11981},[104,105],[24,11983,82],{"className":11984},[152],[24,11986],{"className":11987,"style":111},[110],[24,11989,599],{"className":11990},[115],[24,11992],{"className":11993,"style":111},[110],[24,11995,11997,12001,12067,12070,12110],{"className":11996},[95],[24,11998],{"className":11999,"style":12000},[99],"height:1.1488em;vertical-align:-0.2997em;",[24,12002,12004,12009],{"className":12003},[784],[24,12005,605],{"className":12006,"style":12008},[784,830,12007],"small-op","position:relative;top:0em;",[24,12010,12012],{"className":12011},[700],[24,12013,12015,12058],{"className":12014},[704,705],[24,12016,12018,12055],{"className":12017},[709],[24,12019,12022,12043],{"className":12020,"style":12021},[713],"height:0.8043em;",[24,12023,12025,12028],{"style":12024},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[24,12026],{"className":12027,"style":722},[721],[24,12029,12031],{"className":12030},[726,727,728,729],[24,12032,12034,12037,12040],{"className":12033},[104,729],[24,12035,610],{"className":12036,"style":812},[104,105,729],[24,12038,599],{"className":12039},[115,729],[24,12041,69],{"className":12042},[104,729],[24,12044,12046,12049],{"style":12045},"top:-3.2029em;margin-right:0.05em;",[24,12047],{"className":12048,"style":722},[721],[24,12050,12052],{"className":12051},[726,727,728,729],[24,12053,2614],{"className":12054},[104,105,729],[24,12056,737],{"className":12057},[736],[24,12059,12061],{"className":12060},[709],[24,12062,12065],{"className":12063,"style":12064},[713],"height:0.2997em;",[24,12066],{},[24,12068],{"className":12069,"style":145},[110],[24,12071,12073,12076],{"className":12072},[104],[24,12074,2619],{"className":12075},[104,105],[24,12077,12079],{"className":12078},[700],[24,12080,12082,12102],{"className":12081},[704,705],[24,12083,12085,12099],{"className":12084},[709],[24,12086,12088],{"className":12087,"style":2354},[713],[24,12089,12090,12093],{"style":2357},[24,12091],{"className":12092,"style":722},[721],[24,12094,12096],{"className":12095},[726,727,728,729],[24,12097,610],{"className":12098,"style":812},[104,105,729],[24,12100,737],{"className":12101},[736],[24,12103,12105],{"className":12104},[709],[24,12106,12108],{"className":12107,"style":744},[713],[24,12109],{},[24,12111,12113,12116],{"className":12112},[104],[24,12114,322],{"className":12115},[104,105],[24,12117,12119],{"className":12118},[700],[24,12120,12122],{"className":12121},[704],[24,12123,12125],{"className":12124},[709],[24,12126,12128],{"className":12127,"style":10316},[713],[24,12129,12130,12133],{"style":10271},[24,12131],{"className":12132,"style":722},[721],[24,12134,12136],{"className":12135},[726,727,728,729],[24,12137,610],{"className":12138,"style":812},[104,105,729],", for each ",[24,12141,12143,12160],{"className":12142,"translate":28},[32],[24,12144,12146],{"className":12145},[36],[38,12147,12148],{"xmlns":40},[43,12149,12150,12158],{},[46,12151,12152],{},[650,12153,12154,12156],{},[49,12155,322],{},[49,12157,610],{},[84,12159,10306],{"encoding":86},[24,12161,12163],{"className":12162,"ariaHidden":72},[91],[24,12164,12166,12169],{"className":12165},[95],[24,12167],{"className":12168,"style":10316},[99],[24,12170,12172,12175],{"className":12171},[104],[24,12173,322],{"className":12174},[104,105],[24,12176,12178],{"className":12177},[700],[24,12179,12181],{"className":12180},[704],[24,12182,12184],{"className":12183},[709],[24,12185,12187],{"className":12186,"style":10316},[713],[24,12188,12189,12192],{"style":10271},[24,12190],{"className":12191,"style":722},[721],[24,12193,12195],{"className":12194},[726,727,728,729],[24,12196,610],{"className":12197,"style":812},[104,105,729]," find the corresponding neuron combination ",[24,12200,12202,12226],{"className":12201,"translate":28},[32],[24,12203,12205],{"className":12204},[36],[38,12206,12207],{"xmlns":40},[43,12208,12209,12223],{},[46,12210,12211,12217,12219,12221],{},[577,12212,12213,12215],{},[49,12214,9218],{},[49,12216,610],{},[53,12218,62],{"stretchy":61},[49,12220,322],{},[53,12222,82],{"stretchy":61},[84,12224,12225],{"encoding":86},"N_k(x)",[24,12227,12229],{"className":12228,"ariaHidden":72},[91],[24,12230,12232,12235,12276,12279,12282],{"className":12231},[95],[24,12233],{"className":12234,"style":125},[99],[24,12236,12238,12241],{"className":12237},[104],[24,12239,9218],{"className":12240,"style":6985},[104,105],[24,12242,12244],{"className":12243},[700],[24,12245,12247,12268],{"className":12246},[704,705],[24,12248,12250,12265],{"className":12249},[709],[24,12251,12253],{"className":12252,"style":2354},[713],[24,12254,12256,12259],{"style":12255},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[24,12257],{"className":12258,"style":722},[721],[24,12260,12262],{"className":12261},[726,727,728,729],[24,12263,610],{"className":12264,"style":812},[104,105,729],[24,12266,737],{"className":12267},[736],[24,12269,12271],{"className":12270},[709],[24,12272,12274],{"className":12273,"style":744},[713],[24,12275],{},[24,12277,62],{"className":12278},[133],[24,12280,322],{"className":12281},[104,105],[24,12283,82],{"className":12284},[152]," that approximates it:",[24,12287,12289],{"className":12288,"translate":28},[27],[24,12290,12292,12348],{"className":12291,"translate":28},[32],[24,12293,12295],{"className":12294},[36],[38,12296,12297],{"xmlns":40,"display":41},[43,12298,12299,12345],{},[46,12300,12301,12303,12309,12311,12313,12315,12317,12323,12325,12327],{},[49,12302,369],{"mathvariant":368},[577,12304,12305,12307],{},[49,12306,9218],{},[49,12308,610],{},[53,12310,62],{"stretchy":61},[49,12312,322],{},[53,12314,82],{"stretchy":61},[53,12316,380],{},[650,12318,12319,12321],{},[49,12320,322],{},[49,12322,610],{},[49,12324,369],{"mathvariant":368},[53,12326,393],{},[628,12328,12329,12331],{},[49,12330,9162],{},[46,12332,12333,12335,12337,12339,12341,12343],{},[67,12334,5358],{},[53,12336,62],{"stretchy":61},[49,12338,2614],{},[53,12340,1571],{},[67,12342,76],{},[53,12344,82],{"stretchy":61},[84,12346,12347],{"encoding":86},"  |N_k(x) - x^k| \u003C \\frac{\\epsilon}{2 (m+1)}",[24,12349,12351,12418,12465],{"className":12350,"ariaHidden":72},[91],[24,12352,12354,12357,12360,12400,12403,12406,12409,12412,12415],{"className":12353},[95],[24,12355],{"className":12356,"style":125},[99],[24,12358,369],{"className":12359},[104],[24,12361,12363,12366],{"className":12362},[104],[24,12364,9218],{"className":12365,"style":6985},[104,105],[24,12367,12369],{"className":12368},[700],[24,12370,12372,12392],{"className":12371},[704,705],[24,12373,12375,12389],{"className":12374},[709],[24,12376,12378],{"className":12377,"style":2354},[713],[24,12379,12380,12383],{"style":12255},[24,12381],{"className":12382,"style":722},[721],[24,12384,12386],{"className":12385},[726,727,728,729],[24,12387,610],{"className":12388,"style":812},[104,105,729],[24,12390,737],{"className":12391},[736],[24,12393,12395],{"className":12394},[709],[24,12396,12398],{"className":12397,"style":744},[713],[24,12399],{},[24,12401,62],{"className":12402},[133],[24,12404,322],{"className":12405},[104,105],[24,12407,82],{"className":12408},[152],[24,12410],{"className":12411,"style":450},[110],[24,12413,380],{"className":12414},[454],[24,12416],{"className":12417,"style":450},[110],[24,12419,12421,12424,12453,12456,12459,12462],{"className":12420},[95],[24,12422],{"className":12423,"style":1078},[99],[24,12425,12427,12430],{"className":12426},[104],[24,12428,322],{"className":12429},[104,105],[24,12431,12433],{"className":12432},[700],[24,12434,12436],{"className":12435},[704],[24,12437,12439],{"className":12438},[709],[24,12440,12442],{"className":12441,"style":1044},[713],[24,12443,12444,12447],{"style":1047},[24,12445],{"className":12446,"style":722},[721],[24,12448,12450],{"className":12449},[726,727,728,729],[24,12451,610],{"className":12452,"style":812},[104,105,729],[24,12454,369],{"className":12455},[104],[24,12457],{"className":12458,"style":111},[110],[24,12460,393],{"className":12461},[115],[24,12463],{"className":12464,"style":111},[110],[24,12466,12468,12472],{"className":12467},[95],[24,12469],{"className":12470,"style":12471},[99],"height:2.0436em;vertical-align:-0.936em;",[24,12473,12475,12478,12553],{"className":12474},[104],[24,12476],{"className":12477},[133,891],[24,12479,12481],{"className":12480},[628],[24,12482,12484,12544],{"className":12483},[704,705],[24,12485,12487,12541],{"className":12486},[709],[24,12488,12490,12522,12530],{"className":12489,"style":986},[713],[24,12491,12492,12495],{"style":907},[24,12493],{"className":12494,"style":911},[721],[24,12496,12498,12501,12504,12507,12510,12513,12516,12519],{"className":12497},[104],[24,12499,5358],{"className":12500},[104],[24,12502,62],{"className":12503},[133],[24,12505,2614],{"className":12506},[104,105],[24,12508],{"className":12509,"style":450},[110],[24,12511,1571],{"className":12512},[454],[24,12514],{"className":12515,"style":450},[110],[24,12517,76],{"className":12518},[104],[24,12520,82],{"className":12521},[152],[24,12523,12524,12527],{"style":920},[24,12525],{"className":12526,"style":911},[721],[24,12528],{"className":12529,"style":928},[927],[24,12531,12532,12535],{"style":931},[24,12533],{"className":12534,"style":911},[721],[24,12536,12538],{"className":12537},[104],[24,12539,9162],{"className":12540},[104,105],[24,12542,737],{"className":12543},[736],[24,12545,12547],{"className":12546},[709],[24,12548,12551],{"className":12549,"style":12550},[713],"height:0.936em;",[24,12552],{},[24,12554],{"className":12555},[152,891],[20,12557,12558],{},"Then combine:",[24,12560,12562],{"className":12561,"translate":28},[27],[24,12563,12565,12619],{"className":12564,"translate":28},[32],[24,12566,12568],{"className":12567},[36],[38,12569,12570],{"xmlns":40,"display":41},[43,12571,12572,12616],{},[46,12573,12574,12576,12578,12580,12582,12584,12598,12604,12610,12612,12614],{},[49,12575,9218],{},[53,12577,62],{"stretchy":61},[49,12579,322],{},[53,12581,82],{"stretchy":61},[53,12583,599],{},[601,12585,12586,12588,12596],{},[53,12587,605],{},[46,12589,12590,12592,12594],{},[49,12591,610],{},[53,12593,599],{},[67,12595,69],{},[49,12597,2614],{},[577,12599,12600,12602],{},[49,12601,2619],{},[49,12603,610],{},[577,12605,12606,12608],{},[49,12607,9218],{},[49,12609,610],{},[53,12611,62],{"stretchy":61},[49,12613,322],{},[53,12615,82],{"stretchy":61},[84,12617,12618],{"encoding":86},"  N(x) = \\sum_{k=0}^m a_k N_k(x)",[24,12620,12622,12649],{"className":12621,"ariaHidden":72},[91],[24,12623,12625,12628,12631,12634,12637,12640,12643,12646],{"className":12624},[95],[24,12626],{"className":12627,"style":125},[99],[24,12629,9218],{"className":12630,"style":6985},[104,105],[24,12632,62],{"className":12633},[133],[24,12635,322],{"className":12636},[104,105],[24,12638,82],{"className":12639},[152],[24,12641],{"className":12642,"style":111},[110],[24,12644,599],{"className":12645},[115],[24,12647],{"className":12648,"style":111},[110],[24,12650,12652,12655,12719,12722,12762,12802,12805,12808],{"className":12651},[95],[24,12653],{"className":12654,"style":780},[99],[24,12656,12658],{"className":12657},[784,785],[24,12659,12661,12711],{"className":12660},[704,705],[24,12662,12664,12708],{"className":12663},[709],[24,12665,12667,12687,12697],{"className":12666,"style":795},[713],[24,12668,12669,12672],{"style":798},[24,12670],{"className":12671,"style":802},[721],[24,12673,12675],{"className":12674},[726,727,728,729],[24,12676,12678,12681,12684],{"className":12677},[104,729],[24,12679,610],{"className":12680,"style":812},[104,105,729],[24,12682,599],{"className":12683},[115,729],[24,12685,69],{"className":12686},[104,729],[24,12688,12689,12692],{"style":821},[24,12690],{"className":12691,"style":802},[721],[24,12693,12694],{},[24,12695,605],{"className":12696},[784,830,831],[24,12698,12699,12702],{"style":834},[24,12700],{"className":12701,"style":802},[721],[24,12703,12705],{"className":12704},[726,727,728,729],[24,12706,2614],{"className":12707},[104,105,729],[24,12709,737],{"className":12710},[736],[24,12712,12714],{"className":12713},[709],[24,12715,12717],{"className":12716,"style":856},[713],[24,12718],{},[24,12720],{"className":12721,"style":145},[110],[24,12723,12725,12728],{"className":12724},[104],[24,12726,2619],{"className":12727},[104,105],[24,12729,12731],{"className":12730},[700],[24,12732,12734,12754],{"className":12733},[704,705],[24,12735,12737,12751],{"className":12736},[709],[24,12738,12740],{"className":12739,"style":2354},[713],[24,12741,12742,12745],{"style":2357},[24,12743],{"className":12744,"style":722},[721],[24,12746,12748],{"className":12747},[726,727,728,729],[24,12749,610],{"className":12750,"style":812},[104,105,729],[24,12752,737],{"className":12753},[736],[24,12755,12757],{"className":12756},[709],[24,12758,12760],{"className":12759,"style":744},[713],[24,12761],{},[24,12763,12765,12768],{"className":12764},[104],[24,12766,9218],{"className":12767,"style":6985},[104,105],[24,12769,12771],{"className":12770},[700],[24,12772,12774,12794],{"className":12773},[704,705],[24,12775,12777,12791],{"className":12776},[709],[24,12778,12780],{"className":12779,"style":2354},[713],[24,12781,12782,12785],{"style":12255},[24,12783],{"className":12784,"style":722},[721],[24,12786,12788],{"className":12787},[726,727,728,729],[24,12789,610],{"className":12790,"style":812},[104,105,729],[24,12792,737],{"className":12793},[736],[24,12795,12797],{"className":12796},[709],[24,12798,12800],{"className":12799,"style":744},[713],[24,12801],{},[24,12803,62],{"className":12804},[133],[24,12806,322],{"className":12807},[104,105],[24,12809,82],{"className":12810},[152],[20,12812,12813],{},"Error bound:",[24,12815,12817],{"className":12816,"translate":28},[27],[24,12818,12820,12956],{"className":12819,"translate":28},[32],[24,12821,12823],{"className":12822},[36],[38,12824,12825],{"xmlns":40,"display":41},[43,12826,12827,12953],{},[46,12828,12829,12831,12833,12835,12837,12839,12841,12843,12845,12847,12849,12851,12853,12867,12869,12875,12877,12879,12885,12887,12889,12891,12893,12899,12901,12903,12917,12919,12925,12927,12945,12947,12949,12951],{},[49,12830,369],{"mathvariant":368},[49,12832,9218],{},[53,12834,62],{"stretchy":61},[49,12836,322],{},[53,12838,82],{"stretchy":61},[53,12840,380],{},[49,12842,317],{},[53,12844,62],{"stretchy":61},[49,12846,322],{},[53,12848,82],{"stretchy":61},[49,12850,369],{"mathvariant":368},[53,12852,5919],{},[601,12854,12855,12857,12865],{},[53,12856,605],{},[46,12858,12859,12861,12863],{},[49,12860,610],{},[53,12862,599],{},[67,12864,69],{},[49,12866,2614],{},[49,12868,369],{"mathvariant":368},[577,12870,12871,12873],{},[49,12872,2619],{},[49,12874,610],{},[49,12876,369],{"mathvariant":368},[49,12878,369],{"mathvariant":368},[577,12880,12881,12883],{},[49,12882,9218],{},[49,12884,610],{},[53,12886,62],{"stretchy":61},[49,12888,322],{},[53,12890,82],{"stretchy":61},[53,12892,380],{},[650,12894,12895,12897],{},[49,12896,322],{},[49,12898,610],{},[49,12900,369],{"mathvariant":368},[53,12902,393],{},[601,12904,12905,12907,12915],{},[53,12906,605],{},[46,12908,12909,12911,12913],{},[49,12910,610],{},[53,12912,599],{},[67,12914,69],{},[49,12916,2614],{},[49,12918,369],{"mathvariant":368},[577,12920,12921,12923],{},[49,12922,2619],{},[49,12924,610],{},[49,12926,369],{"mathvariant":368},[628,12928,12929,12931],{},[49,12930,9162],{},[46,12932,12933,12935,12937,12939,12941,12943],{},[67,12934,5358],{},[53,12936,62],{"stretchy":61},[49,12938,2614],{},[53,12940,1571],{},[67,12942,76],{},[53,12944,82],{"stretchy":61},[53,12946,393],{},[49,12948,9162],{},[49,12950,5572],{"mathvariant":368},[67,12952,5358],{},[84,12954,12955],{"encoding":86},"  |N(x) - P(x)| \\le \\sum_{k=0}^m |a_k||N_k(x)-x^k| \u003C \\sum_{k=0}^m |a_k| \\frac{\\epsilon}{2(m+1)} \u003C \\epsilon\u002F2",[24,12957,12959,12989,13019,13197,13244,13455],{"className":12958,"ariaHidden":72},[91],[24,12960,12962,12965,12968,12971,12974,12977,12980,12983,12986],{"className":12961},[95],[24,12963],{"className":12964,"style":125},[99],[24,12966,369],{"className":12967},[104],[24,12969,9218],{"className":12970,"style":6985},[104,105],[24,12972,62],{"className":12973},[133],[24,12975,322],{"className":12976},[104,105],[24,12978,82],{"className":12979},[152],[24,12981],{"className":12982,"style":450},[110],[24,12984,380],{"className":12985},[454],[24,12987],{"className":12988,"style":450},[110],[24,12990,12992,12995,12998,13001,13004,13007,13010,13013,13016],{"className":12991},[95],[24,12993],{"className":12994,"style":125},[99],[24,12996,317],{"className":12997,"style":340},[104,105],[24,12999,62],{"className":13000},[133],[24,13002,322],{"className":13003},[104,105],[24,13005,82],{"className":13006},[152],[24,13008,369],{"className":13009},[104],[24,13011],{"className":13012,"style":111},[110],[24,13014,5919],{"className":13015},[115],[24,13017],{"className":13018,"style":111},[110],[24,13020,13022,13025,13089,13092,13095,13135,13139,13179,13182,13185,13188,13191,13194],{"className":13021},[95],[24,13023],{"className":13024,"style":780},[99],[24,13026,13028],{"className":13027},[784,785],[24,13029,13031,13081],{"className":13030},[704,705],[24,13032,13034,13078],{"className":13033},[709],[24,13035,13037,13057,13067],{"className":13036,"style":795},[713],[24,13038,13039,13042],{"style":798},[24,13040],{"className":13041,"style":802},[721],[24,13043,13045],{"className":13044},[726,727,728,729],[24,13046,13048,13051,13054],{"className":13047},[104,729],[24,13049,610],{"className":13050,"style":812},[104,105,729],[24,13052,599],{"className":13053},[115,729],[24,13055,69],{"className":13056},[104,729],[24,13058,13059,13062],{"style":821},[24,13060],{"className":13061,"style":802},[721],[24,13063,13064],{},[24,13065,605],{"className":13066},[784,830,831],[24,13068,13069,13072],{"style":834},[24,13070],{"className":13071,"style":802},[721],[24,13073,13075],{"className":13074},[726,727,728,729],[24,13076,2614],{"className":13077},[104,105,729],[24,13079,737],{"className":13080},[736],[24,13082,13084],{"className":13083},[709],[24,13085,13087],{"className":13086,"style":856},[713],[24,13088],{},[24,13090],{"className":13091,"style":145},[110],[24,13093,369],{"className":13094},[104],[24,13096,13098,13101],{"className":13097},[104],[24,13099,2619],{"className":13100},[104,105],[24,13102,13104],{"className":13103},[700],[24,13105,13107,13127],{"className":13106},[704,705],[24,13108,13110,13124],{"className":13109},[709],[24,13111,13113],{"className":13112,"style":2354},[713],[24,13114,13115,13118],{"style":2357},[24,13116],{"className":13117,"style":722},[721],[24,13119,13121],{"className":13120},[726,727,728,729],[24,13122,610],{"className":13123,"style":812},[104,105,729],[24,13125,737],{"className":13126},[736],[24,13128,13130],{"className":13129},[709],[24,13131,13133],{"className":13132,"style":744},[713],[24,13134],{},[24,13136,13138],{"className":13137},[104],"∣∣",[24,13140,13142,13145],{"className":13141},[104],[24,13143,9218],{"className":13144,"style":6985},[104,105],[24,13146,13148],{"className":13147},[700],[24,13149,13151,13171],{"className":13150},[704,705],[24,13152,13154,13168],{"className":13153},[709],[24,13155,13157],{"className":13156,"style":2354},[713],[24,13158,13159,13162],{"style":12255},[24,13160],{"className":13161,"style":722},[721],[24,13163,13165],{"className":13164},[726,727,728,729],[24,13166,610],{"className":13167,"style":812},[104,105,729],[24,13169,737],{"className":13170},[736],[24,13172,13174],{"className":13173},[709],[24,13175,13177],{"className":13176,"style":744},[713],[24,13178],{},[24,13180,62],{"className":13181},[133],[24,13183,322],{"className":13184},[104,105],[24,13186,82],{"className":13187},[152],[24,13189],{"className":13190,"style":450},[110],[24,13192,380],{"className":13193},[454],[24,13195],{"className":13196,"style":450},[110],[24,13198,13200,13203,13232,13235,13238,13241],{"className":13199},[95],[24,13201],{"className":13202,"style":1078},[99],[24,13204,13206,13209],{"className":13205},[104],[24,13207,322],{"className":13208},[104,105],[24,13210,13212],{"className":13211},[700],[24,13213,13215],{"className":13214},[704],[24,13216,13218],{"className":13217},[709],[24,13219,13221],{"className":13220,"style":1044},[713],[24,13222,13223,13226],{"style":1047},[24,13224],{"className":13225,"style":722},[721],[24,13227,13229],{"className":13228},[726,727,728,729],[24,13230,610],{"className":13231,"style":812},[104,105,729],[24,13233,369],{"className":13234},[104],[24,13236],{"className":13237,"style":111},[110],[24,13239,393],{"className":13240},[115],[24,13242],{"className":13243,"style":111},[110],[24,13245,13247,13250,13314,13317,13320,13360,13363,13446,13449,13452],{"className":13246},[95],[24,13248],{"className":13249,"style":780},[99],[24,13251,13253],{"className":13252},[784,785],[24,13254,13256,13306],{"className":13255},[704,705],[24,13257,13259,13303],{"className":13258},[709],[24,13260,13262,13282,13292],{"className":13261,"style":795},[713],[24,13263,13264,13267],{"style":798},[24,13265],{"className":13266,"style":802},[721],[24,13268,13270],{"className":13269},[726,727,728,729],[24,13271,13273,13276,13279],{"className":13272},[104,729],[24,13274,610],{"className":13275,"style":812},[104,105,729],[24,13277,599],{"className":13278},[115,729],[24,13280,69],{"className":13281},[104,729],[24,13283,13284,13287],{"style":821},[24,13285],{"className":13286,"style":802},[721],[24,13288,13289],{},[24,13290,605],{"className":13291},[784,830,831],[24,13293,13294,13297],{"style":834},[24,13295],{"className":13296,"style":802},[721],[24,13298,13300],{"className":13299},[726,727,728,729],[24,13301,2614],{"className":13302},[104,105,729],[24,13304,737],{"className":13305},[736],[24,13307,13309],{"className":13308},[709],[24,13310,13312],{"className":13311,"style":856},[713],[24,13313],{},[24,13315],{"className":13316,"style":145},[110],[24,13318,369],{"className":13319},[104],[24,13321,13323,13326],{"className":13322},[104],[24,13324,2619],{"className":13325},[104,105],[24,13327,13329],{"className":13328},[700],[24,13330,13332,13352],{"className":13331},[704,705],[24,13333,13335,13349],{"className":13334},[709],[24,13336,13338],{"className":13337,"style":2354},[713],[24,13339,13340,13343],{"style":2357},[24,13341],{"className":13342,"style":722},[721],[24,13344,13346],{"className":13345},[726,727,728,729],[24,13347,610],{"className":13348,"style":812},[104,105,729],[24,13350,737],{"className":13351},[736],[24,13353,13355],{"className":13354},[709],[24,13356,13358],{"className":13357,"style":744},[713],[24,13359],{},[24,13361,369],{"className":13362},[104],[24,13364,13366,13369,13443],{"className":13365},[104],[24,13367],{"className":13368},[133,891],[24,13370,13372],{"className":13371},[628],[24,13373,13375,13435],{"className":13374},[704,705],[24,13376,13378,13432],{"className":13377},[709],[24,13379,13381,13413,13421],{"className":13380,"style":986},[713],[24,13382,13383,13386],{"style":907},[24,13384],{"className":13385,"style":911},[721],[24,13387,13389,13392,13395,13398,13401,13404,13407,13410],{"className":13388},[104],[24,13390,5358],{"className":13391},[104],[24,13393,62],{"className":13394},[133],[24,13396,2614],{"className":13397},[104,105],[24,13399],{"className":13400,"style":450},[110],[24,13402,1571],{"className":13403},[454],[24,13405],{"className":13406,"style":450},[110],[24,13408,76],{"className":13409},[104],[24,13411,82],{"className":13412},[152],[24,13414,13415,13418],{"style":920},[24,13416],{"className":13417,"style":911},[721],[24,13419],{"className":13420,"style":928},[927],[24,13422,13423,13426],{"style":931},[24,13424],{"className":13425,"style":911},[721],[24,13427,13429],{"className":13428},[104],[24,13430,9162],{"className":13431},[104,105],[24,13433,737],{"className":13434},[736],[24,13436,13438],{"className":13437},[709],[24,13439,13441],{"className":13440,"style":12550},[713],[24,13442],{},[24,13444],{"className":13445},[152,891],[24,13447],{"className":13448,"style":111},[110],[24,13450,393],{"className":13451},[115],[24,13453],{"className":13454,"style":111},[110],[24,13456,13458,13461,13464],{"className":13457},[95],[24,13459],{"className":13460,"style":125},[99],[24,13462,9162],{"className":13463},[104,105],[24,13465,10012],{"className":13466},[104],[20,13468,13469],{},[555,13470,13471],{},"Step 4: Triangle inequality completes the approximation",[20,13473,13474],{},"We have two sources of error:",[13476,13477,13478,13602],"ol",{},[13479,13480,13481,13601],"li",{},[24,13482,13484,13526],{"className":13483,"translate":28},[32],[24,13485,13487],{"className":13486},[36],[38,13488,13489],{"xmlns":40},[43,13490,13491,13523],{},[46,13492,13493,13495,13497,13499,13501,13503,13505,13507,13509,13511,13513,13515,13517,13519,13521],{},[49,13494,369],{"mathvariant":368},[49,13496,51],{},[53,13498,62],{"stretchy":61},[49,13500,322],{},[53,13502,82],{"stretchy":61},[53,13504,380],{},[49,13506,317],{},[53,13508,62],{"stretchy":61},[49,13510,322],{},[53,13512,82],{"stretchy":61},[49,13514,369],{"mathvariant":368},[53,13516,393],{},[49,13518,9162],{},[49,13520,5572],{"mathvariant":368},[67,13522,5358],{},[84,13524,13525],{"encoding":86},"|f(x) - P(x)| \u003C \\epsilon\u002F2",[24,13527,13529,13559,13589],{"className":13528,"ariaHidden":72},[91],[24,13530,13532,13535,13538,13541,13544,13547,13550,13553,13556],{"className":13531},[95],[24,13533],{"className":13534,"style":125},[99],[24,13536,369],{"className":13537},[104],[24,13539,51],{"className":13540,"style":106},[104,105],[24,13542,62],{"className":13543},[133],[24,13545,322],{"className":13546},[104,105],[24,13548,82],{"className":13549},[152],[24,13551],{"className":13552,"style":450},[110],[24,13554,380],{"className":13555},[454],[24,13557],{"className":13558,"style":450},[110],[24,13560,13562,13565,13568,13571,13574,13577,13580,13583,13586],{"className":13561},[95],[24,13563],{"className":13564,"style":125},[99],[24,13566,317],{"className":13567,"style":340},[104,105],[24,13569,62],{"className":13570},[133],[24,13572,322],{"className":13573},[104,105],[24,13575,82],{"className":13576},[152],[24,13578,369],{"className":13579},[104],[24,13581],{"className":13582,"style":111},[110],[24,13584,393],{"className":13585},[115],[24,13587],{"className":13588,"style":111},[110],[24,13590,13592,13595,13598],{"className":13591},[95],[24,13593],{"className":13594,"style":125},[99],[24,13596,9162],{"className":13597},[104,105],[24,13599,10012],{"className":13600},[104]," (Weierstrass theorem)",[13479,13603,13604,13724],{},[24,13605,13607,13649],{"className":13606,"translate":28},[32],[24,13608,13610],{"className":13609},[36],[38,13611,13612],{"xmlns":40},[43,13613,13614,13646],{},[46,13615,13616,13618,13620,13622,13624,13626,13628,13630,13632,13634,13636,13638,13640,13642,13644],{},[49,13617,369],{"mathvariant":368},[49,13619,317],{},[53,13621,62],{"stretchy":61},[49,13623,322],{},[53,13625,82],{"stretchy":61},[53,13627,380],{},[49,13629,9218],{},[53,13631,62],{"stretchy":61},[49,13633,322],{},[53,13635,82],{"stretchy":61},[49,13637,369],{"mathvariant":368},[53,13639,393],{},[49,13641,9162],{},[49,13643,5572],{"mathvariant":368},[67,13645,5358],{},[84,13647,13648],{"encoding":86},"|P(x) - N(x)| \u003C \\epsilon\u002F2",[24,13650,13652,13682,13712],{"className":13651,"ariaHidden":72},[91],[24,13653,13655,13658,13661,13664,13667,13670,13673,13676,13679],{"className":13654},[95],[24,13656],{"className":13657,"style":125},[99],[24,13659,369],{"className":13660},[104],[24,13662,317],{"className":13663,"style":340},[104,105],[24,13665,62],{"className":13666},[133],[24,13668,322],{"className":13669},[104,105],[24,13671,82],{"className":13672},[152],[24,13674],{"className":13675,"style":450},[110],[24,13677,380],{"className":13678},[454],[24,13680],{"className":13681,"style":450},[110],[24,13683,13685,13688,13691,13694,13697,13700,13703,13706,13709],{"className":13684},[95],[24,13686],{"className":13687,"style":125},[99],[24,13689,9218],{"className":13690,"style":6985},[104,105],[24,13692,62],{"className":13693},[133],[24,13695,322],{"className":13696},[104,105],[24,13698,82],{"className":13699},[152],[24,13701,369],{"className":13702},[104],[24,13704],{"className":13705,"style":111},[110],[24,13707,393],{"className":13708},[115],[24,13710],{"className":13711,"style":111},[110],[24,13713,13715,13718,13721],{"className":13714},[95],[24,13716],{"className":13717,"style":125},[99],[24,13719,9162],{"className":13720},[104,105],[24,13722,10012],{"className":13723},[104]," (neuron approximation of polynomials)",[20,13726,13727],{},"By the triangle inequality:",[24,13729,13731],{"className":13730,"translate":28},[27],[24,13732,13734,13836],{"className":13733,"translate":28},[32],[24,13735,13737],{"className":13736},[36],[38,13738,13739],{"xmlns":40,"display":41},[43,13740,13741,13833],{},[46,13742,13743,13745,13747,13749,13751,13753,13755,13757,13759,13761,13763,13765,13767,13769,13771,13773,13775,13777,13779,13781,13783,13785,13787,13789,13791,13793,13795,13797,13799,13801,13803,13805,13807,13809,13811,13813,13815,13817,13819,13821,13823,13825,13827,13829,13831],{},[49,13744,369],{"mathvariant":368},[49,13746,51],{},[53,13748,62],{"stretchy":61},[49,13750,322],{},[53,13752,82],{"stretchy":61},[53,13754,380],{},[49,13756,9218],{},[53,13758,62],{"stretchy":61},[49,13760,322],{},[53,13762,82],{"stretchy":61},[49,13764,369],{"mathvariant":368},[53,13766,5919],{},[49,13768,369],{"mathvariant":368},[49,13770,51],{},[53,13772,62],{"stretchy":61},[49,13774,322],{},[53,13776,82],{"stretchy":61},[53,13778,380],{},[49,13780,317],{},[53,13782,62],{"stretchy":61},[49,13784,322],{},[53,13786,82],{"stretchy":61},[49,13788,369],{"mathvariant":368},[53,13790,1571],{},[49,13792,369],{"mathvariant":368},[49,13794,317],{},[53,13796,62],{"stretchy":61},[49,13798,322],{},[53,13800,82],{"stretchy":61},[53,13802,380],{},[49,13804,9218],{},[53,13806,62],{"stretchy":61},[49,13808,322],{},[53,13810,82],{"stretchy":61},[49,13812,369],{"mathvariant":368},[53,13814,393],{},[49,13816,9162],{},[49,13818,5572],{"mathvariant":368},[67,13820,5358],{},[53,13822,1571],{},[49,13824,9162],{},[49,13826,5572],{"mathvariant":368},[67,13828,5358],{},[53,13830,599],{},[49,13832,9162],{},[84,13834,13835],{"encoding":86},"  |f(x) - N(x)| \\le |f(x)-P(x)| + |P(x)-N(x)| \u003C \\epsilon\u002F2 + \\epsilon\u002F2 = \\epsilon",[24,13837,13839,13869,13899,13929,13959,13989,14019,14040,14061],{"className":13838,"ariaHidden":72},[91],[24,13840,13842,13845,13848,13851,13854,13857,13860,13863,13866],{"className":13841},[95],[24,13843],{"className":13844,"style":125},[99],[24,13846,369],{"className":13847},[104],[24,13849,51],{"className":13850,"style":106},[104,105],[24,13852,62],{"className":13853},[133],[24,13855,322],{"className":13856},[104,105],[24,13858,82],{"className":13859},[152],[24,13861],{"className":13862,"style":450},[110],[24,13864,380],{"className":13865},[454],[24,13867],{"className":13868,"style":450},[110],[24,13870,13872,13875,13878,13881,13884,13887,13890,13893,13896],{"className":13871},[95],[24,13873],{"className":13874,"style":125},[99],[24,13876,9218],{"className":13877,"style":6985},[104,105],[24,13879,62],{"className":13880},[133],[24,13882,322],{"className":13883},[104,105],[24,13885,82],{"className":13886},[152],[24,13888,369],{"className":13889},[104],[24,13891],{"className":13892,"style":111},[110],[24,13894,5919],{"className":13895},[115],[24,13897],{"className":13898,"style":111},[110],[24,13900,13902,13905,13908,13911,13914,13917,13920,13923,13926],{"className":13901},[95],[24,13903],{"className":13904,"style":125},[99],[24,13906,369],{"className":13907},[104],[24,13909,51],{"className":13910,"style":106},[104,105],[24,13912,62],{"className":13913},[133],[24,13915,322],{"className":13916},[104,105],[24,13918,82],{"className":13919},[152],[24,13921],{"className":13922,"style":450},[110],[24,13924,380],{"className":13925},[454],[24,13927],{"className":13928,"style":450},[110],[24,13930,13932,13935,13938,13941,13944,13947,13950,13953,13956],{"className":13931},[95],[24,13933],{"className":13934,"style":125},[99],[24,13936,317],{"className":13937,"style":340},[104,105],[24,13939,62],{"className":13940},[133],[24,13942,322],{"className":13943},[104,105],[24,13945,82],{"className":13946},[152],[24,13948,369],{"className":13949},[104],[24,13951],{"className":13952,"style":450},[110],[24,13954,1571],{"className":13955},[454],[24,13957],{"className":13958,"style":450},[110],[24,13960,13962,13965,13968,13971,13974,13977,13980,13983,13986],{"className":13961},[95],[24,13963],{"className":13964,"style":125},[99],[24,13966,369],{"className":13967},[104],[24,13969,317],{"className":13970,"style":340},[104,105],[24,13972,62],{"className":13973},[133],[24,13975,322],{"className":13976},[104,105],[24,13978,82],{"className":13979},[152],[24,13981],{"className":13982,"style":450},[110],[24,13984,380],{"className":13985},[454],[24,13987],{"className":13988,"style":450},[110],[24,13990,13992,13995,13998,14001,14004,14007,14010,14013,14016],{"className":13991},[95],[24,13993],{"className":13994,"style":125},[99],[24,13996,9218],{"className":13997,"style":6985},[104,105],[24,13999,62],{"className":14000},[133],[24,14002,322],{"className":14003},[104,105],[24,14005,82],{"className":14006},[152],[24,14008,369],{"className":14009},[104],[24,14011],{"className":14012,"style":111},[110],[24,14014,393],{"className":14015},[115],[24,14017],{"className":14018,"style":111},[110],[24,14020,14022,14025,14028,14031,14034,14037],{"className":14021},[95],[24,14023],{"className":14024,"style":125},[99],[24,14026,9162],{"className":14027},[104,105],[24,14029,10012],{"className":14030},[104],[24,14032],{"className":14033,"style":450},[110],[24,14035,1571],{"className":14036},[454],[24,14038],{"className":14039,"style":450},[110],[24,14041,14043,14046,14049,14052,14055,14058],{"className":14042},[95],[24,14044],{"className":14045,"style":125},[99],[24,14047,9162],{"className":14048},[104,105],[24,14050,10012],{"className":14051},[104],[24,14053],{"className":14054,"style":111},[110],[24,14056,599],{"className":14057},[115],[24,14059],{"className":14060,"style":111},[110],[24,14062,14064,14067],{"className":14063},[95],[24,14065],{"className":14066,"style":1374},[99],[24,14068,9162],{"className":14069},[104,105],[20,14071,14072],{},"QED.",[20,14074,14075],{},"In other words: the nonlinear units of a neural network act as an \"adjustable polynomial basis\", and a single layer can combine them into any polynomial, thereby approximating any continuous function.",[10,14077,14079],{"id":14078},"uat-and-the-machine-learning-perspective","UAT and the Machine Learning Perspective",[20,14081,14082],{},"This proof is actually quite interesting:",[20,14084,14085],{},"The Bernstein polynomial essentially expresses",[24,14087,14089],{"className":14088,"translate":28},[27],[24,14090,14092,14188],{"className":14091,"translate":28},[32],[24,14093,14095],{"className":14094},[36],[38,14096,14097],{"xmlns":40,"display":41},[43,14098,14099,14185],{},[46,14100,14101,14103,14105,14107,14109,14111,14117,14119,14121,14133,14135],{},[49,14102,51],{},[53,14104,62],{"stretchy":61},[49,14106,322],{},[53,14108,82],{"stretchy":61},[53,14110,4529],{},[8659,14112,14113,14115],{},[53,14114,605],{},[49,14116,610],{},[49,14118,51],{},[620,14120,622],{},[46,14122,14123,14125,14131],{},[53,14124,62],{"fence":72},[628,14126,14127,14129],{},[49,14128,610],{},[49,14130,584],{},[53,14132,82],{"fence":72},[53,14134,7271],{},[8659,14136,14137,14182],{},[8659,14138,14139,14179],{},[46,14140,14141,14153,14159,14161,14163,14165,14167],{},[46,14142,14143,14145,14151],{},[53,14144,62],{"fence":72},[628,14146,14147,14149],{"linethickness":642},[49,14148,584],{},[49,14150,610],{},[53,14152,82],{"fence":72},[650,14154,14155,14157],{},[49,14156,322],{},[49,14158,610],{},[53,14160,62],{"stretchy":61},[67,14162,76],{},[53,14164,380],{},[49,14166,322],{},[650,14168,14169,14171],{},[53,14170,82],{"stretchy":61},[46,14172,14173,14175,14177],{},[49,14174,584],{},[53,14176,380],{},[49,14178,610],{},[53,14180,14181],{"stretchy":72},"⏟",[620,14183,14184],{},"base function",[84,14186,14187],{"encoding":86},"f(x)\n\\approx\n\\sum_k\nf\\!\\left(\\frac{k}{n}\\right)\n\\cdot\n\\underbrace{\n\\binom{n}{k}\nx^k(1-x)^{n-k}\n}_{\\text{base function}}",[24,14189,14191,14218,14367],{"className":14190,"ariaHidden":72},[91],[24,14192,14194,14197,14200,14203,14206,14209,14212,14215],{"className":14193},[95],[24,14195],{"className":14196,"style":125},[99],[24,14198,51],{"className":14199,"style":106},[104,105],[24,14201,62],{"className":14202},[133],[24,14204,322],{"className":14205},[104,105],[24,14207,82],{"className":14208},[152],[24,14210],{"className":14211,"style":111},[110],[24,14213,4529],{"className":14214},[115],[24,14216],{"className":14217,"style":111},[110],[24,14219,14221,14225,14269,14272,14275,14278,14281,14358,14361,14364],{"className":14220},[95],[24,14222],{"className":14223,"style":14224},[99],"height:2.7521em;vertical-align:-1.3021em;",[24,14226,14228],{"className":14227},[784,785],[24,14229,14231,14261],{"className":14230},[704,705],[24,14232,14234,14258],{"className":14233},[709],[24,14235,14237,14248],{"className":14236,"style":11650},[713],[24,14238,14239,14242],{"style":798},[24,14240],{"className":14241,"style":802},[721],[24,14243,14245],{"className":14244},[726,727,728,729],[24,14246,610],{"className":14247,"style":812},[104,105,729],[24,14249,14250,14253],{"style":821},[24,14251],{"className":14252,"style":802},[721],[24,14254,14255],{},[24,14256,605],{"className":14257},[784,830,831],[24,14259,737],{"className":14260},[736],[24,14262,14264],{"className":14263},[709],[24,14265,14267],{"className":14266,"style":856},[713],[24,14268],{},[24,14270],{"className":14271,"style":145},[110],[24,14273,51],{"className":14274,"style":106},[104,105],[24,14276],{"className":14277,"style":868},[110],[24,14279],{"className":14280,"style":145},[110],[24,14282,14284,14290,14352],{"className":14283},[875],[24,14285,14287],{"className":14286,"style":880},[133,879],[24,14288,62],{"className":14289},[884,728],[24,14291,14293,14296,14349],{"className":14292},[104],[24,14294],{"className":14295},[133,891],[24,14297,14299],{"className":14298},[628],[24,14300,14302,14341],{"className":14301},[704,705],[24,14303,14305,14338],{"className":14304},[709],[24,14306,14308,14319,14327],{"className":14307,"style":904},[713],[24,14309,14310,14313],{"style":907},[24,14311],{"className":14312,"style":911},[721],[24,14314,14316],{"className":14315},[104],[24,14317,584],{"className":14318},[104,105],[24,14320,14321,14324],{"style":920},[24,14322],{"className":14323,"style":911},[721],[24,14325],{"className":14326,"style":928},[927],[24,14328,14329,14332],{"style":931},[24,14330],{"className":14331,"style":911},[721],[24,14333,14335],{"className":14334},[104],[24,14336,610],{"className":14337,"style":812},[104,105],[24,14339,737],{"className":14340},[736],[24,14342,14344],{"className":14343},[709],[24,14345,14347],{"className":14346,"style":950},[713],[24,14348],{},[24,14350],{"className":14351},[152,891],[24,14353,14355],{"className":14354,"style":880},[152,879],[24,14356,82],{"className":14357},[884,728],[24,14359],{"className":14360,"style":450},[110],[24,14362,7271],{"className":14363},[454],[24,14365],{"className":14366,"style":450},[110],[24,14368,14370,14374],{"className":14369},[95],[24,14371],{"className":14372,"style":14373},[99],"height:3.7341em;vertical-align:-2.2841em;",[24,14375,14377],{"className":14376},[875,8659],[24,14378,14380,14639],{"className":14379},[704,705],[24,14381,14383,14636],{"className":14382},[709],[24,14384,14387,14406],{"className":14385,"style":14386},[713],"height:1.45em;",[24,14388,14390,14394],{"style":14389},"top:-1.1659em;",[24,14391],{"className":14392,"style":14393},[721],"height:3.45em;",[24,14395,14397],{"className":14396},[726,727,728,729],[24,14398,14400],{"className":14399},[104,729],[24,14401,14403],{"className":14402},[104,9942,729],[24,14404,14184],{"className":14405},[104,729],[24,14407,14409,14412],{"style":14408},"top:-3.45em;",[24,14410],{"className":14411,"style":14393},[721],[24,14413,14415],{"className":14414},[875,8659],[24,14416,14418,14627],{"className":14417},[704,705],[24,14419,14421,14624],{"className":14420},[709],[24,14422,14424,14471],{"className":14423,"style":14386},[713],[24,14425,14429,14432],{"className":14426,"style":14428},[14427],"svg-align","top:-1.852em;",[24,14430],{"className":14431,"style":14393},[721],[24,14433,14437,14451,14461],{"className":14434,"style":14436},[14435],"stretchy","height:0.548em;min-width:1.6em;",[24,14438,14442],{"className":14439,"style":14441},[14440],"brace-left","height:0.548em;",[5743,14443,14448],{"xmlns":5745,"width":14444,"height":14445,"viewBox":14446,"preserveAspectRatio":14447},"400em","0.548em","0 0 400000 548","xMinYMin slice",[5749,14449],{"d":14450},"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",[24,14452,14455],{"className":14453,"style":14441},[14454],"brace-center",[5743,14456,14458],{"xmlns":5745,"width":14444,"height":14445,"viewBox":14446,"preserveAspectRatio":14457},"xMidYMin slice",[5749,14459],{"d":14460},"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 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20-1h17z",[24,14472,14473,14476],{"style":14408},[24,14474],{"className":14475,"style":14393},[721],[24,14477,14479,14539,14568,14571,14574,14577,14580,14583,14586],{"className":14478},[104],[24,14480,14482,14488,14533],{"className":14481},[104],[24,14483,14485],{"className":14484,"style":880},[133,879],[24,14486,62],{"className":14487},[884,728],[24,14489,14491],{"className":14490},[628],[24,14492,14494,14525],{"className":14493},[704,705],[24,14495,14497,14522],{"className":14496},[709],[24,14498,14500,14511],{"className":14499,"style":986},[713],[24,14501,14502,14505],{"style":907},[24,14503],{"className":14504,"style":911},[721],[24,14506,14508],{"className":14507},[104],[24,14509,610],{"className":14510,"style":812},[104,105],[24,14512,14513,14516],{"style":931},[24,14514],{"className":14515,"style":911},[721],[24,14517,14519],{"className":14518},[104],[24,14520,584],{"className":14521},[104,105],[24,14523,737],{"className":14524},[736],[24,14526,14528],{"className":14527},[709],[24,14529,14531],{"className":14530,"style":950},[713],[24,14532],{},[24,14534,14536],{"className":14535,"style":880},[152,879],[24,14537,82],{"className":14538},[884,728],[24,14540,14542,14545],{"className":14541},[104],[24,14543,322],{"className":14544},[104,105],[24,14546,14548],{"className":14547},[700],[24,14549,14551],{"className":14550},[704],[24,14552,14554],{"className":14553},[709],[24,14555,14557],{"className":14556,"style":1044},[713],[24,14558,14559,14562],{"style":1047},[24,14560],{"className":14561,"style":722},[721],[24,14563,14565],{"className":14564},[726,727,728,729],[24,14566,610],{"className":14567,"style":812},[104,105,729],[24,14569,62],{"className":14570},[133],[24,14572,76],{"className":14573},[104],[24,14575],{"className":14576,"style":450},[110],[24,14578,380],{"className":14579},[454],[24,14581],{"className":14582,"style":450},[110],[24,14584,322],{"className":14585},[104,105],[24,14587,14589,14592],{"className":14588},[152],[24,14590,82],{"className":14591},[152],[24,14593,14595],{"className":14594},[700],[24,14596,14598],{"className":14597},[704],[24,14599,14601],{"className":14600},[709],[24,14602,14604],{"className":14603,"style":1044},[713],[24,14605,14606,14609],{"style":1047},[24,14607],{"className":14608,"style":722},[721],[24,14610,14612],{"className":14611},[726,727,728,729],[24,14613,14615,14618,14621],{"className":14614},[104,729],[24,14616,584],{"className":14617},[104,105,729],[24,14619,380],{"className":14620},[454,729],[24,14622,610],{"className":14623,"style":812},[104,105,729],[24,14625,737],{"className":14626},[736],[24,14628,14630],{"className":14629},[709],[24,14631,14634],{"className":14632,"style":14633},[713],"height:1.598em;",[24,14635],{},[24,14637,737],{"className":14638},[736],[24,14640,14642],{"className":14641},[709],[24,14643,14646],{"className":14644,"style":14645},[713],"height:2.2841em;",[24,14647],{},[20,14649,14650],{},"This closely resembles a neural network",[24,14652,14654],{"className":14653,"translate":28},[27],[24,14655,14657,14716],{"className":14656,"translate":28},[32],[24,14658,14660],{"className":14659},[36],[38,14661,14662],{"xmlns":40,"display":41},[43,14663,14664,14713],{},[46,14665,14666,14668,14670,14672,14674,14676,14682,14688,14690,14692,14694,14701,14703,14705,14711],{},[49,14667,51],{},[53,14669,62],{"stretchy":61},[49,14671,322],{},[53,14673,82],{"stretchy":61},[53,14675,4529],{},[8659,14677,14678,14680],{},[53,14679,605],{},[49,14681,2606],{},[577,14683,14684,14686],{},[49,14685,2619],{},[49,14687,2606],{},[53,14689,7271],{"separator":72},[49,14691,9250],{},[53,14693,62],{"stretchy":61},[577,14695,14696,14699],{},[49,14697,14698],{},"w",[49,14700,2606],{},[49,14702,322],{},[53,14704,1571],{},[577,14706,14707,14709],{},[49,14708,9070],{},[49,14710,2606],{},[53,14712,82],{"stretchy":61},[84,14714,14715],{"encoding":86},"f(x)\n\\approx\n\\sum_i\na_i·\\sigma(w_i x+b_i)",[24,14717,14719,14746,14905],{"className":14718,"ariaHidden":72},[91],[24,14720,14722,14725,14728,14731,14734,14737,14740,14743],{"className":14721},[95],[24,14723],{"className":14724,"style":125},[99],[24,14726,51],{"className":14727,"style":106},[104,105],[24,14729,62],{"className":14730},[133],[24,14732,322],{"className":14733},[104,105],[24,14735,82],{"className":14736},[152],[24,14738],{"className":14739,"style":111},[110],[24,14741,4529],{"className":14742},[115],[24,14744],{"className":14745,"style":111},[110],[24,14747,14749,14752,14796,14799,14839,14842,14845,14848,14851,14893,14896,14899,14902],{"className":14748},[95],[24,14750],{"className":14751,"style":11637},[99],[24,14753,14755],{"className":14754},[784,785],[24,14756,14758,14788],{"className":14757},[704,705],[24,14759,14761,14785],{"className":14760},[709],[24,14762,14764,14775],{"className":14763,"style":11650},[713],[24,14765,14766,14769],{"style":9327},[24,14767],{"className":14768,"style":802},[721],[24,14770,14772],{"className":14771},[726,727,728,729],[24,14773,2606],{"className":14774},[104,105,729],[24,14776,14777,14780],{"style":821},[24,14778],{"className":14779,"style":802},[721],[24,14781,14782],{},[24,14783,605],{"className":14784},[784,830,831],[24,14786,737],{"className":14787},[736],[24,14789,14791],{"className":14790},[709],[24,14792,14794],{"className":14793,"style":9379},[713],[24,14795],{},[24,14797],{"className":14798,"style":145},[110],[24,14800,14802,14805],{"className":14801},[104],[24,14803,2619],{"className":14804},[104,105],[24,14806,14808],{"className":14807},[700],[24,14809,14811,14831],{"className":14810},[704,705],[24,14812,14814,14828],{"className":14813},[709],[24,14815,14817],{"className":14816,"style":9404},[713],[24,14818,14819,14822],{"style":2357},[24,14820],{"className":14821,"style":722},[721],[24,14823,14825],{"className":14824},[726,727,728,729],[24,14826,2606],{"className":14827},[104,105,729],[24,14829,737],{"className":14830},[736],[24,14832,14834],{"className":14833},[709],[24,14835,14837],{"className":14836,"style":744},[713],[24,14838],{},[24,14840,7271],{"className":14841},[141],[24,14843],{"className":14844,"style":145},[110],[24,14846,9250],{"className":14847,"style":5287},[104,105],[24,14849,62],{"className":14850},[133],[24,14852,14854,14858],{"className":14853},[104],[24,14855,14698],{"className":14856,"style":14857},[104,105],"margin-right:0.0269em;",[24,14859,14861],{"className":14860},[700],[24,14862,14864,14885],{"className":14863},[704,705],[24,14865,14867,14882],{"className":14866},[709],[24,14868,14870],{"className":14869,"style":9404},[713],[24,14871,14873,14876],{"style":14872},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[24,14874],{"className":14875,"style":722},[721],[24,14877,14879],{"className":14878},[726,727,728,729],[24,14880,2606],{"className":14881},[104,105,729],[24,14883,737],{"className":14884},[736],[24,14886,14888],{"className":14887},[709],[24,14889,14891],{"className":14890,"style":744},[713],[24,14892],{},[24,14894,322],{"className":14895},[104,105],[24,14897],{"className":14898,"style":450},[110],[24,14900,1571],{"className":14901},[454],[24,14903],{"className":14904,"style":450},[110],[24,14906,14908,14911,14951],{"className":14907},[95],[24,14909],{"className":14910,"style":125},[99],[24,14912,14914,14917],{"className":14913},[104],[24,14915,9070],{"className":14916},[104,105],[24,14918,14920],{"className":14919},[700],[24,14921,14923,14943],{"className":14922},[704,705],[24,14924,14926,14940],{"className":14925},[709],[24,14927,14929],{"className":14928,"style":9404},[713],[24,14930,14931,14934],{"style":2357},[24,14932],{"className":14933,"style":722},[721],[24,14935,14937],{"className":14936},[726,727,728,729],[24,14938,2606],{"className":14939},[104,105,729],[24,14941,737],{"className":14942},[736],[24,14944,14946],{"className":14945},[709],[24,14947,14949],{"className":14948,"style":744},[713],[24,14950],{},[24,14952,82],{"className":14953},[152],[20,14955,14956],{},"The difference is only:",[14958,14959,14960,15105],"ul",{},[13479,14961,14962,14963],{},"Weierstrass uses polynomial bases ",[24,14964,14966,15004],{"className":14965,"translate":28},[32],[24,14967,14969],{"className":14968},[36],[38,14970,14971],{"xmlns":40},[43,14972,14973,15001],{},[46,14974,14975,14981,14983,14985,14987,14989],{},[650,14976,14977,14979],{},[49,14978,322],{},[49,14980,610],{},[53,14982,62],{"stretchy":61},[67,14984,76],{},[53,14986,380],{},[49,14988,322],{},[650,14990,14991,14993],{},[53,14992,82],{"stretchy":61},[46,14994,14995,14997,14999],{},[49,14996,584],{},[53,14998,380],{},[49,15000,610],{},[84,15002,15003],{"encoding":86},"x^k(1-x)^{n-k}",[24,15005,15007,15058],{"className":15006,"ariaHidden":72},[91],[24,15008,15010,15014,15043,15046,15049,15052,15055],{"className":15009},[95],[24,15011],{"className":15012,"style":15013},[99],"height:1.0991em;vertical-align:-0.25em;",[24,15015,15017,15020],{"className":15016},[104],[24,15018,322],{"className":15019},[104,105],[24,15021,15023],{"className":15022},[700],[24,15024,15026],{"className":15025},[704],[24,15027,15029],{"className":15028},[709],[24,15030,15032],{"className":15031,"style":10316},[713],[24,15033,15034,15037],{"style":10271},[24,15035],{"className":15036,"style":722},[721],[24,15038,15040],{"className":15039},[726,727,728,729],[24,15041,610],{"className":15042,"style":812},[104,105,729],[24,15044,62],{"className":15045},[133],[24,15047,76],{"className":15048},[104],[24,15050],{"className":15051,"style":450},[110],[24,15053,380],{"className":15054},[454],[24,15056],{"className":15057,"style":450},[110],[24,15059,15061,15064,15067],{"className":15060},[95],[24,15062],{"className":15063,"style":15013},[99],[24,15065,322],{"className":15066},[104,105],[24,15068,15070,15073],{"className":15069},[152],[24,15071,82],{"className":15072},[152],[24,15074,15076],{"className":15075},[700],[24,15077,15079],{"className":15078},[704],[24,15080,15082],{"className":15081},[709],[24,15083,15085],{"className":15084,"style":10316},[713],[24,15086,15087,15090],{"style":10271},[24,15088],{"className":15089,"style":722},[721],[24,15091,15093],{"className":15092},[726,727,728,729],[24,15094,15096,15099,15102],{"className":15095},[104,729],[24,15097,584],{"className":15098},[104,105,729],[24,15100,380],{"className":15101},[454,729],[24,15103,610],{"className":15104,"style":812},[104,105,729],[13479,15106,15107,15108],{},"Neural networks use activation function bases ",[24,15109,15111,15137],{"className":15110,"translate":28},[32],[24,15112,15114],{"className":15113},[36],[38,15115,15116],{"xmlns":40},[43,15117,15118,15134],{},[46,15119,15120,15122,15124,15126,15128,15130,15132],{},[49,15121,9250],{},[53,15123,62],{"stretchy":61},[49,15125,14698],{},[49,15127,322],{},[53,15129,1571],{},[49,15131,9070],{},[53,15133,82],{"stretchy":61},[84,15135,15136],{"encoding":86},"\\sigma(wx+b)",[24,15138,15140,15167],{"className":15139,"ariaHidden":72},[91],[24,15141,15143,15146,15149,15152,15155,15158,15161,15164],{"className":15142},[95],[24,15144],{"className":15145,"style":125},[99],[24,15147,9250],{"className":15148,"style":5287},[104,105],[24,15150,62],{"className":15151},[133],[24,15153,14698],{"className":15154,"style":14857},[104,105],[24,15156,322],{"className":15157},[104,105],[24,15159],{"className":15160,"style":450},[110],[24,15162,1571],{"className":15163},[454],[24,15165],{"className":15166,"style":450},[110],[24,15168,15170,15173,15176],{"className":15169},[95],[24,15171],{"className":15172,"style":125},[99],[24,15174,9070],{"className":15175},[104,105],[24,15177,82],{"className":15178},[152],[20,15180,15181],{},"Thus from a modern perspective, the UAT theorem can be understood as a generalization of the Weierstrass theorem to \"neural network basis functions\". Weierstrass says \"the polynomial basis is dense\", while UAT says \"the neuron basis is also dense\".",{"title":15183,"searchDepth":15184,"depth":15184,"links":15185},"",2,[15186,15191,15192],{"id":12,"depth":15184,"text":13,"children":15187},[15188,15190],{"id":17,"depth":15189,"text":18},3,{"id":5051,"depth":15189,"text":5052},{"id":9034,"depth":15184,"text":9035},{"id":14078,"depth":15184,"text":14079},null,false,true,"md",{},"2026-06-06","\u002Fblog\u002F2026\u002F2026-06-06-proof-of-the-uat-by-weierstrass-theory",{"title":5,"description":15183},"blog\u002F2026\u002F2026-06-06-proof-of-the-uat-by-weierstrass-theory","The core purpose of UAT (Universal Approximation Theorem) is that neural networks can approximate any continuous function, and continuous functions can be approximated by polynomials (Weierstrass theorem).",[15204,15205],"machine-learning","mathematics","t240_-OZm75Qa18MY9UMMwWM7rRIuvAEqwX62pZtC4c",1781192430089]