以下公式来源于South Caolina
1.$\frac{1}{1-x}$
$$ \begin{aligned}\frac{1}{1-x} &=\quad1+x+x^{2}+x^{3}+x^{4}+\ldots \\&=\quad\sum_{n=0}^{\infty} x^{n}\end{aligned} $$
当为几何级数时。只需将$x$视为$r$
$x\in (-1,1)$
2. $e^x$
$$ \begin{aligned}e^{x}\quad &=\quad 1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\ldots \\&=\quad\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\end{aligned} $$
3.$\cos{x}$
$$ \begin{aligned}\cos x\quad &=\quad1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\frac{x^{8}}{8 !}-\ldots \\&=\quad\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}\end{aligned} $$
4.$\sin{x}$
$$ \begin{aligned}\sin x \quad &=\quad x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\frac{x^{9}}{9 !}-\ldots \\&=\quad \sum_{n=1}^{\infty}(-1)^{(n-1)} \frac{x^{2 n-1}}{(2 n-1) !} \stackrel{\text { or }}{=} \sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}\end{aligned} $$
5.$\ln{(1+x)}$
$$ \begin{aligned}\ln (1+x)\quad &=\quad x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\frac{x^{5}}{5}-\ldots \\&=\quad\sum_{n=1}^{\infty}(-1)^{(n-1)} \frac{x^{n}}{n} \stackrel{\text { or }}{=} \sum_{n=1}^{\infty}(-1)^{n+1} \frac{x^{n}}{n}\end{aligned} $$
6.$\tan ^{-1} x$
$$ \begin{aligned}\tan ^{-1} x \quad &= \quad x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\frac{x^{9}}{9}-\ldots \\&= \quad\sum_{n=1}^{\infty}(-1)^{(n-1)} \frac{x^{2 n-1}}{2 n-1} \stackrel{\text { or }}{=} \sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{2 n+1}\end{aligned} $$