数学

收敛域除特别标注外均为 $|x| < 1$ ：

$\frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 + \dots + (-1)^n x^{2n} + o(x^{2n}) = \sum_{n=0}^\infty (-1)^n x^{2n}$
$\frac{1}{1-x^2} = 1 + x^2 + x^4 + x^6 + \dots + x^{2n} + o(x^{2n}) = \sum_{n=0}^\infty x^{2n}$
$\frac{1}{1+x} = 1 - x + x^2 - x^3 + \dots + (-1)^n x^n + o(x^n)$
$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots + x^n + o(x^n)$
$\frac{1}{(1-x)^2} = 1 + 2x + 3x^2 + \dots + n x^{n-1} + o(x^{n-1})$ （求导法可得）
$\frac{1}{(1+x)^2} = 1 - 2x + 3x^2 - \dots + (-1)^{n-1} n x^{n-1} + o(x^{n-1})$
$\sqrt{1+x} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 + o(x^3)$
$\frac{1}{\sqrt{1+x}} = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 + o(x^3)$
$\frac{1}{\sqrt{1-x^2}} = 1 + \frac{1}{2}x^2 + \frac{3}{8}x^4 + o(x^4)$

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!} + o(x^n)$ $\quad (x \in \mathbb{R})$
$a^x = 1 + x\ln a + \frac{(x\ln a)^2}{2!} + o(x^2)$
$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots + (-1)^{n-1}\frac{x^n}{n} + o(x^n)$ $\quad (-1 < x \le 1)$
$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} + \dots + (-1)^n\frac{x^{2n+1}}{(2n+1)!} + o(x^{2n+2})$ $\quad (x \in \mathbb{R})$
$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \dots + (-1)^n\frac{x^{2n}}{(2n)!} + o(x^{2n+1})$ $\quad (x \in \mathbb{R})$
$\tan x = x + \frac{1}{3}x^3 + \frac{2}{15}x^5 + o(x^5)$
$\arcsin x = x + \frac{1}{6}x^3 + \frac{3}{40}x^5 + o(x^5)$ $\quad (|x| \le 1)$
$\arctan x = x - \frac{1}{3}x^3 + \frac{1}{5}x^5 + \dots + (-1)^n\frac{x^{2n+1}}{2n+1} + o(x^{2n+2})$ $\quad (|x| \le 1)$
$\sec x = 1 + \frac{1}{2}x^2 + \frac{5}{24}x^4 + o(x^4)$

$(1+x)^{\frac{1}{x}} = e\left(1 - \frac{1}{2}x + \frac{11}{24}x^2 - \frac{7}{16}x^3\right) + o(x^3)$
$\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + o(x^5)$
$\cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + o(x^4)$

当 $x \to 0$ 时：