常用的麦克劳林级数展开式(泰勒展开式)

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

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

$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots +\frac{(-1{)}^n}{(2n+1)!}{x}^{2n+1}+o(x^{2n+1})$

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

$tanx=x+\frac{1}{3}{x}^3+\frac{1}{5}{x}^5+\cdots +\frac{1}{2n+1}{x}^{2n+1}+o(x^{2n+1})$
$$\begin{array}{rlrlrlrlrlrlrlrlrlrlrlrlrlrlrlr}推导：& (\tan x-x)\sim \frac{1}{3}{x}^3\sim (x-\arctan x)& & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & (x-\sin x)\sim \frac{1}{6}{x}^3\sim (\arcsin x-x)& & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & \alpha \sim \beta \Rightarrow \alpha =\beta +o(\beta )& & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & 得\tan x=x+\frac{1}{3}{x}^3+o(x^3)& & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & 同理\arctan x，\arcsin x& & & & & & & & & & & & & & & & & & & & & & & & & & & & & \end{array}$$

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

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

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

$\arcsin x=x+\frac{1}{2}\times \frac{x^3}{3}+\frac{1\times 3}{2\times 4}\times \frac{x^5}{5}+o(x^5)=x+\frac{x^3}{6}+o(x^3)$

$\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}+\cdots +\frac{(-1{)}^n}{2n+1}{x}^{2n+1}+o(x^{2n+1})$