：12ChapterInfiniteSeriesSec.4 PowerSeriesRepresentationforFunctions
Chapter 12 Infinite Series Sec.4 Power Series Representation for Functions
S4 Power SeriesRepresentationforFunctionsI.Introduction801Z>(-1<x<1)Y(-1<x<1)-x1-xn=0n=0880f(x)=Zf(x)=Ea,(x-xo)"anxn=0n=0Questions1. Under what condition1 can a function be representedby a power series?2. If a function can be represented by a power series,what is a, then ?3.Is the power series unique?
§4 Power Series Representation for Functions I. Introduction ( 1 1) 1 1 0 − − = = x x x n n n n f (x) an (x x ) 0 0 = − = n n n f x a x = = 0 ( ) Questions 2. If a function can be represented by a power series, what is then ? n a 3. Is the power series unique? 1. Under what condition can a function be represented by a power series? ( 1 1) 1 1 0 = − − = x x x n n
S4 Power SeriesRepresentationforFunctionsI.IntroductionTaylorTheoremLet f be a function whose(n + 1)stderivetivef(n+1)(x)existsfor e achx in an ope ninte rvalI containing Xo. The n,for e achx in If"(xo)f(x)= f(x)+ f'(x,)(x-x,)+ 12!g(n) (xo)x-xo)" +R,(x)n!n+l)()(x-xo)"+1Eis betweenxand x,R,(x):(n + 1)!
§4 Power Series Representation for Functions n n x x n f x x x f x f x f x f x x x ( ) ! ( ) ( ) 2! ( ) ( ) ( ) ( )( ) 0 0 ( ) 2 0 0 0 0 0 + + − − = + − + R (x) + n I. Introduction 1 0 ( 1) ( ) ( 1)! ( ) ( ) + + − + = n n n x x n f R x Taylor Theorem for each in an openinterval containing .Then,for each in I, Let be a function whose ( 1)st derivetive ( ) exists 0 ( 1) x I x x f n f x n+ + between and 0 is x x
S4 Power SeriesRepresentationforFunctionsI.Introductionxf(x) = f(x)+ f'(x)(x-x)2!x.-x)" +R;(x)Taylor seriesn!Whether the Taylor series converges to f(x)?lim Sn+i(x) = f(x) 台 lim[f(x)- Sn+i(x)]= 0n-→>800r(n+1) ()n+1台 lim R(x) = 0 ← lim=0-Xo(n + 1)!n->00n->00
§4 Power Series Representation for Functions n n x x n f x x x f x f x f x f x x x ( ) ! ( ) ( ) 2! ( ) ( ) ( ) ( )( ) 0 0 ( ) 2 0 0 0 0 0 + + − − = + − + R (x) ++ n Whether the Taylor series converges to ? f (x) lim ( ) ( ) Sn 1 x f x n + = → lim[ ( ) − +1 ( )] = 0 → f x Sn x n lim ( ) = 0 → R x n ( ) 0 ( 1)! ( ) lim 1 0 ( 1) − = + + + → n n n x x n f I. Introduction Taylor series
S4 PowerSeriesRepresentationforFunctionsIl.Taylor'sTheoremTheoremLetf be a function with derivatives of all ordersin theinterval(x, - S,x, + ). The Taylor seriesf"(xo)x-x。f(xo)+ f'(xo)(x -xo)+2!re pre se ntsthe function f on this intervalif and only if()(x-x,)n+1 = 0lim R,(x) = lim(n+1)!n-→>0n→0
§4 Power Series Representation for Functions representsthe function o n this intervalif and only if ( ) 2! ( ) ( ) ( )( ) interval( , ).The Taylor series Let be a function with derivatives o f all ordersin the 2 0 0 0 0 0 0 0 f x x f x f x f x x x x x f − + + − + − + ( ) 0 ( 1)! ( ) lim ( ) lim 1 0 ( 1) − = + = + + → → n n n n n x x n f R x II. Taylor’s Theorem Theorem