几个重要的展开式：

$$e^x = 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} + o(x^n)$$

$$\sin x = x – \frac{x^3}{3!} + \cdots + (-1)^{n-1} \frac{x^{2n-1}}{(2n-1)!} + o(x^{2n})$$

$$\cos x = 1 – \frac{x^2}{2!} + \cdots + (-1)^n \frac{x^{2n}}{(2n)!} + o(x^{2n+1})$$

$$(1+x)^m = 1 + mx + \frac{m(m-1)}{2!}x^2 + \cdots + \frac{m(m-1) \cdots (m-n+1)}{n!}x^n + o(x^n)$$

$$\ln (1+x) = x – \frac{x^2}{2!} + \cdots + (-1)^{n-1} \frac{x^n}{n} + o(x^n)$$

补充一个$f(x) = \frac{1}{\sqrt{1+x}}$的麦克劳林公式：

$$\frac{1}{\sqrt{1+x}} = 1 – \frac{1}{2}x + \frac{1 \times 3}{2 \times 4}x^2 + \cdots +(-1)^n \frac{(2n-1)!!}{(2n)!!}x^n + o(x^n)$$

同理：

$$\frac{1}{\sqrt{1-x}} = 1 + \frac{1}{2}x + \frac{1 \times 3}{2 \times 4}x^2 + \cdots + \frac{(2n-1)!!}{(2n)!!}x^n + o(x^n)$$

补充一个：

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