数学分析(十四)

f ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + f ′ ′ ( x 0 ) 2 ! ( x − x 0 ) 2 + ⋯ + f ( n ) ( x 0 ) n ! ( x − x 0 ) n + R n ( x ) , ( 1 ) \begin{aligned} f(x)=f\left(x_{0}\right)+ & f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)+\frac{f^{\prime \prime}\left(x_{0}\right)}{2 !}\left(x-x_{0}\right)^{2}+\cdots+ \frac{f^{(n)}\left(x_{0}\right)}{n !}\left(x-x_{0}\right)^{n}+R_{n}(x), \quad\quad(1) \end{aligned} f(x)=f(x0​)+​f′(x0​)(x−x0​)+2!f′′(x0​)​(x−x0​)2+⋯+n!f(n)(x0​)​(x−x0​)n+Rn​(x),(1)​

这里 R n ( x ) R_{n}(x) Rn​(x) 为拉格朗日型余项

R n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( x − x 0 ) n + 1 , ( 2 ) R_{n}(x)=\frac{f^{(n+1)}(\xi)}{(n+1) !}\left(x-x_{0}\right)^{n+1}, \quad\quad(2) Rn​(x)=(n+1)!f