如何证明泰勒公式?
By Long Luo
挖坑
\[ \begin{aligned} f(x) & = f(a) + f(x) - f(a) \\ & = f(a) + \int_a^x f'(t) \mathrm{d}t \\ & = f(a) - \int_a^x f'(t) \mathrm{d}(x-t) \\ & = f(a) - f'(t)(x - t)|_a^x - \frac12 \int_a^x f''(t) \mathrm{d}[(x-t)^2] \\ & = f(a) + f'(a)(x - a) + {f''(a) \over 2}(x - a)^2 - {1 \over 3!} \int_a^x f'''(t) \mathrm{d}[(x-t)]^3 \end{aligned} \]
不断重复这一过程，可知当 f 的 n+1 阶导数连续时，有：
\[ f(x) = \sum_{k=0}^n {f^{(k)}(a) \over k!}(x-a)^k + R_n(x) \]
其中积分余项 为：
\[ R_n(x) = {1 \over n!} \int_a^x f^{(n + 1)}(t)(x - t)^n \mathrm{d}t \]