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【正文】 ortheLagrangianrem (951)typeform 00x=,get 2()(0) ()nnfxf xx xrx=++++… , (952)Athispoint,(1) (1)1 11 () ()()(1)! (1)!n nn nn f f xrx xn nξ θ+ ++ ++= =+ + (01θ).That(952)typeform ulashowsthatnyfunction()fxaslongasuntilthe1n+derivative,ncanbequaltoapolynom ialandarem 209NO .8O O OPrincipleofmathematicalnalysi209()2(0)0)()(0)(0)2! !n nf ffxf fx x xn′′=++ ++ +… … (953),isito()fxfortheSum functions?IftheorderMaclurinseris(953)thefirst 1n+item sandfor1()nSx+,whic ()21 (0)0)()(0)(0)2! !n nn f fSxf fx x xn+ ′′=++ ++…Then,thesries(953)convergestothefunction()fxtheconditons1lim ()()nnsxfx+→∞ =.NotingMaclurinform ula(952)andtheMaclurinseris(953)therelationshipbetwentheknown1() ()()n nfxSxrx+=+Thus,when ()0nrx=Ther,1() ()nfxSx+= 1lim ()()nnsxfx+→∞ =,Unitsm ust()0nrx=.ThisindicatesthatheMaclurinseris(953)to()fxandfunctionastheMaclurinform ula(952)oftherm ainderterm ()0nrx→(when→∞).Inthisway,wegtafunction()fxthepowerserisexpansion:() ()0 0) 0)() (0)(0)! !n nn nnf ffx xf fx
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