Taylor's theorem

Exercise

1.

Show that the following MacLaurin expansion.

(a)

$${\cos{x} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots + (-1)^{n}\frac{x^{2n}}{(2n)!} + \cdots,\ (-\infty < x < \infty)}$$

(b)

$${\log(1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \cdots + (-1)^{n-1}\frac{x^{n}}{n} + \cdots , \ (-1 < x \leq 1))}$$

(c)

$${(1+x)^{\alpha} = 1 + \frac{\alpha}{1!}x + \frac{\alpha(\alpha-1)}{2!}x^{2} + \cdots + \frac{\alpha(\alpha-1)\cdots(\alpha -n+1)}{n!}x^{n} + \cdots}$$
where， $(-1 < x < 1)$

2.

Find the following limit using Landau's small o．

(a)

$${\lim_{x \rightarrow 0}\frac{\log{(1+x)}}{x}}$$

(b)

$${\lim_{x \rightarrow 0}\frac{x - \sin{x}}{x^3}}$$

(c)

$${\lim_{x \rightarrow 0}\frac{e^{x} - 1 - x}{x^2}}$$

(d)

$${\lim_{x \rightarrow \infty}\frac{x^{\alpha}}{e^{x}}}$$