《高等数学》课程电子教案（PPT课件）Chapter 12 Infinite series Sec 4 Power Series Representation for Functions

：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

S4 Power Series RepresentationforFunctionsIl.Taylor'sTheoremWe can prove that the representation is unique.f("(x)n = 0,1,2,...an!f"(0)If x= 0, f(x)= f(O)+ f(O)x+2!Maclaurinseries

§4 Power Series Representation for Functions II. Taylor’s Theorem Maclaurin series We can prove that the representation is unique. 0,1,2, ! ( ) 0 ( ) = n = n f x a n n +  = = +  + 2 0 2! (0) If 0, ( ) (0) (0) x f x f x f f x

S4 Power SeriesRepresentationforFunctionsIl.Findpowerseriesforfunctionsl.MaclaurinseriesSteps: (1) Find f(")(x) and f(")(O),n = 0,1,2,..Write out the Maclaurin series and findits radius(2) If lim R, = 0 or| (")(x)|≤ M,n-→0Then the series converges to f(x)r(n) (0)f"(0)f(x) = f(0)+ f'(0)x -2!n!

§4 Power Series Representation for Functions II. Find power series for functions 1. Maclaurin series Steps: ( ) (2) If lim 0 or ( ) , n n n R f x M →  =  Then the series converges to ( ). f x ++ +  = +  + n n x n f x f f x f f x ! (0) 2! (0) ( ) (0) (0) ( ) 2 (1) Find f (n) (x) and f (n) (0),n = 0,1,2,  Write out the Maclaurin series and find its radius

S4 Power SeriesRepresentationforFunctionsIl.Find powerseriesforfunctionsE.g.1Find the power series in x of f(x) = e*.Solution:f(n)(x)=e*, f(n)(O)=1.(n = 0,1,2,..)11I+x+R=+802!n!elte5n+1n+1R,(x)Eisbetweenxand0n→ 0(n→8)So we haveex=1+x+: xE(-8,+8)2!n!

§4 Power Series Representation for Functions II. Find power series for functions E.g.1 Solution: Find the power series in of ( ) . x x f x e = ( ) , (n) x f x = e (0) 1. ( 0,1,2, ) f (n) = n =  + + ++ x n + n x x ! 1 2! 1 1 2 So we have ( , ) ! 1 2! 1 1 2 = + + + + x + x  −  +  n e x x x  n  1 ( 1)! ( ) + + = n n x n e R x  1 ( 1)! + +  n x x n e R = + → 0(n → )  is between x and 0

S4 Power SeriesRepresentationforFunctionsIl.Find powerseriesforfunctionsE.g.2 Find the power series in x of f(x) = sinx.n元n元Solution:f(n)(0) = sinf(n)(x) = sin(x +22f(2n+1)(0)=(-1)",(n= 0,1,2,.):. f(2n) (0) = 0,F2n+11R=+0(-1)x3!5!(2n+1)!n元AndLf(n)(x)1sin(x≤1-22n+1x E(-8,+8). sinx =(2n + 1)!

§4 Power Series Representation for Functions II. Find power series for functions E.g.2 Solution: R = + ), 2 ( ) sin( ( )  = + n f x x n , 2 (0) sin ( )  = n f n (0) 0, (2 )  = n f (0) ( 1) , (2n 1) n f = − + (n = 0,1,2, ) ( )( ) n A nd f x = ) 2 sin(  + n x  1  + +  = − + − + − + (2 1)! ( 1) 5! 1 3! 1 sin 2 1 3 5 n x x x x x n n x(−,+) Find the power series in of ( ) sin . x f x x =  + + − + − + − + (2 1)! ( 1) 5! 1 3! 1 2 1 3 5 n x x x x n n

S4 Power Series Representation forFunctionsIl.Findpowerseriesforfunctions2.IndirectMethodWe can find the representation of a function bysubstitution, four arithmetic operation, term byterm differentiation and integration, for the repre-sentation of a function is unique

§4 Power Series Representation for Functions II. Find power series for functions 2. Indirect Method We can find the representation of a function by substitution, four arithmetic operation, term by term differentiation and integration, for the repre￾sentation of a function is unique