例2将f(x)=sinx展开成x的幂级数 解∫((x)=sn(x+), f(0=sin f2(0)=0,f(n-1(0)=(-1)",(n=0,2,…) 且/(x)=smx+ n70 ≤1x∈(-0,+∞) 2 2n+1 ∴sinx=x-,x3+,x3-…+(-1) 5! (2n+1)! x∈(-∞,+o)例 2 将f (x) = sin x展开成x的幂级数. 解 ), 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, ) ( ) = ( ) f x 且 n ) 2 sin(  + n x  1 x  (−,+ )  + +  = − + − + − + (2 1)! ( 1) 5!1 3!1 sin 2 1 3 5 nx x x x x n nx(−,+ )