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2.7 Taylor's theorem

1.

(a) $\displaystyle{f^{(n)}(x) = \cos{(x + \frac{n\pi}{2})}}$より

$\displaystyle \vert R_{n}\vert = \vert\frac{f^{(n)}(\theta x)}{n!}x^{n}\vert \l... ...rt^{n} \leq \frac{1}{n!}\vert x\vert^{n} \rightarrow 0 (n \rightarrow \infty)$

(b) $\displaystyle{f^{(n)}(x) = (-1)^{n-1} (1 + x)^{-n}}$より

$\displaystyle \vert R_{n}\vert = \vert\frac{f^{(n)}(\theta x)}{n!}x^{n}\vert \l... ...\theta x)^{-n}}{n!}\vert\vert x\vert^{n} \rightarrow 0 (n \rightarrow \infty)$

(c) $\displaystyle{f^{(n)}(x) = \alpha \cdot (\alpha - 1) \cdots (\alpha - n + 1)(1 + x)^{\alpha -n}}$より

$\displaystyle \vert R_{n}\vert = \vert\frac{f^{(n)}(\theta x)}{n!}x^{n}\vert \l... ...x)^{\alpha -n}}{n!}\vert\vert x\vert^{n} \rightarrow 0 (n \rightarrow \infty)$

2.

(a) 1(b) $\displaystyle{\frac{1}{6}}$(c) $\displaystyle{\frac{1}{2}}$(d) 0

3. omitted