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标题: 证明题 [打印本页]

作者: quantum 时间: 2014-8-6 10:23
标题: 证明题
这个怎么证明

IMG_20140806_095757_0.jpg (194.06 KB, 下载次数: 148)

作者: castelu 时间: 2014-8-6 21:37
$$\left\{ \begin{array}{l}
f\left( a+h \right) = f\left( a \right) + hf'\left( a \right) + \cdots + \frac{h^{n+1}}{\left( n+1 \right)!}f^{\left( n+1 \right)}\left( a \right) + \frac{h^{n+2}}{\left( n+2 \right)!}f^{\left( n+2 \right)}\left( a \right) + o\left( h^{n + 2} \right),\left( h \rightarrow 0 \right) \\
f\left( a+h \right) = f\left( a \right) + hf'\left( a \right) + \cdots + \frac{h^{n+1}}{\left( n+1 \right)!}f^{\left( n+1 \right)}\left( a + \theta h \right),\left( 0<\theta<1 \right) \\
\end{array} \right.$$
$$\frac{f^{\left( n+1 \right)}\left( a+\theta h \right) - f^{\left( n+1 \right)}\left( a \right)}{\theta h}\theta  = \frac{f^{\left( n+2 \right)}\left( a \right)}{n+2} + o\left( 1 \right)$$
$$f^{\left( n+2 \right)}\left( a \right)\lim\limits_{h \rightarrow 0} \theta  = f^{\left( n+2 \right)}\left( a \right)\lim\limits_{h \rightarrow 0} \frac{1}{n+2}$$
$$\lim\limits_{h \rightarrow 0} \theta  = \frac{1}{n+2}$$

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