庄定理21(在点x的泰勒级数在U(x)内收 敛于∫(x)分在U(x0)内lmRn(x)=0 n→ 证明必要性设f(x)能展开为泰勒级数, /()=2( (x-xo)+rn(x) i=0 i EF:R, (x)=f(x)-suni(x),. lim Sui (x)=f(x) lim r,(x)=limf(x)-sm1(x)=0; n→0 上页定 理 2 f (x)在 点x0的泰勒级数,在 ( ) U x0 内 收 敛 于 f (x)在 ( ) U x0 内lim ( ) = 0 → Rn x n . 证明 必要性 ( ) ( ) ! ( ) ( ) 0 0 0 ( ) x x R x i f x f x n i n i i =  − + =  ( ) ( ) ( ), Rn x = f x − sn+1 x 设f (x)能展开为泰勒级数, lim ( ) ( ) sn 1 x f x n + = →   = → lim R (x) n n lim[ ( ) ( )] f x sn 1 x n + → − = 0;