Taylor Formula

在此篇文章中有一些泰勒展开式，主要用在极限求法上，可以稍微记忆

$$e^{x}=\sum_{n = 0}^{\infty}\frac{x^{n}}{n!}=1 + x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots+\frac{x^{n}}{n!}+\cdots$$

$$\ln(1 + x)=\sum_{n = 1}^{\infty}(-1)^{n - 1}\frac{x^{n}}{n}=x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \cdots + (-1)^{n - 1}\frac{x^{n}}{n} + \cdots \quad (-1 < x \leq 1)$$

$$\sin x=\sum_{k = 0}^{\infty}(-1)^{k}\frac{x^{2k + 1}}{(2k + 1)!}=x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \cdots + (-1)^{k}\frac{x^{2k + 1}}{(2k + 1)!} + \cdots$$

$$\cos x=\sum_{k = 0}^{\infty}(-1)^{k}\frac{x^{2k}}{(2k)!}=1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \cdots + (-1)^{k}\frac{x^{2k}}{(2k)!} + \cdots$$

$$\tan x = x + \frac{1}{3}x^{3} + \frac{2}{15}x^{5} + \cdots\quad (|x| < \frac{\pi}{2})$$

$$\cot x=\frac{1}{x}-\frac{x}{3}-\frac{x^{3}}{45}-\cdots\quad (0 < |x| < \pi)$$

$$\sec x = 1 + \frac{x^{2}}{2} + \frac{5x^{4}}{24} + \frac{61x^{6}}{720} + \cdots = \sum_{n = 0}^{\infty} \frac{E_{2n}}{(2n)!}x^{2n} \quad (|x| < \frac{\pi}{2})$$

$$\csc x = \frac{1}{x} + \frac{x}{6} + \frac{7x^{3}}{360} + \frac{31x^{5}}{15120} + \cdots \quad (0 < |x| < \pi)$$

$$(1 + x)^{\alpha}=\sum_{n = 0}^{\infty}\frac{\alpha(\alpha - 1)\cdots(\alpha - n + 1)}{n!}x^{n}=1 + \alpha x + \frac{\alpha(\alpha - 1)}{2!}x^{2} + \cdots + \frac{\alpha(\alpha - 1)\cdots(\alpha - n + 1)}{n!}x^{n} + \cdots \quad \text{(收敛区间依}\alpha\text{而定)}$$

$$\arctan x=\sum_{n = 0}^{\infty} \frac{(-1)^{n}}{2n + 1}x^{2n + 1}=x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \cdots + \frac{(-1)^{n}}{2n + 1}x^{2n + 1} + \cdots\quad (|x| < 1)$$

$$\arcsin x=\sum_{n = 0}^{\infty} \frac{(2n)!}{4^{n}(n!)^{2}(2n + 1)}x^{2n + 1}=x + \frac{1}{6}x^{3} + \frac{3}{40}x^{5} + \cdots + \frac{(2n)!}{4^{n}(n!)^{2}(2n + 1)}x^{2n + 1} + \cdots\quad (|x| < 1)$$

$$\arccos x=\frac{\pi}{2}-\sum_{n = 0}^{\infty} \frac{(2n)!}{(2n + 1)4^{n}(n!)^{2}}x^{2n + 1}= \frac{\pi}{2}-x - \frac{x^{3}}{6} - \frac{3x^{5}}{40} - \cdots \quad (-1 \leq x \leq 1)$$

$$\sinh x=\sum_{n = 0}^{\infty} \frac{x^{2n + 1}}{(2n + 1)!}=x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \cdots + \frac{x^{2n + 1}}{(2n + 1)!} + \cdots$$

$$\cosh x=\sum_{n = 0}^{\infty} \frac{x^{2n}}{(2n)!}=1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \cdots + \frac{x^{2n}}{(2n)!} + \cdots$$

$$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\left(x - \frac{x^{3}}{3} + \frac{x^{5}}{10} - \frac{x^{7}}{42} + \cdots\right)$$

$$\text{Si}(x)=x - \frac{x^{3}}{3\times3!} + \frac{x^{5}}{5\times5!} - \frac{x^{7}}{7\times7!} + \cdots$$

$$\text{Ci}(x)=\gamma + \ln x - \frac{x^{2}}{2\times2!} + \frac{x^{4}}{4\times4!} - \frac{x^{6}}{6\times6!} + \cdots$$