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楼主 |
发表于 2020-7-17 13:47
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本帖最后由 elim 于 2020-7-16 22:55 编辑
Taylor 定理: 设开区间\(\,I\ni a,\;f\in\mathscr{C}^{k+1}(I),\underset{\,}{\;}\)则
\(\small f(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\cdots+\frac{f^{(k)}(a)}{k!}(x-a)^k+{\displaystyle\int_a^x}\frac{f^{(k+1)}(t)}{k!}(x-t)^kdt\)
证:\(\small\;f(x)=f(a)-{\small\displaystyle\int_a^x} f'(t)d(x-t)\overset{(\dagger)}{=}f(a)+\frac{f'(a)}{1!}(x-a)+\underset{\,}{\small\displaystyle\int_a^x\frac{f''(t)}{1!}}(x-t)dt\)
\(\qquad\cdots \overset{(\ddagger)}{=}\small f(a)+\frac{f'(a)}{1!}(x-a)+\cdots+\frac{f^{(k)}(a)}{k!}(x-a)^k+{\small\displaystyle\int_a^x\frac{f^{(k+1)}(t)}{k!}}(x-t)^kdt.\)
注记:\(\small(\dagger),\;\;(\ddagger)\) 依次表示“分部积分”及“以此类推”. |
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