泰勒展开式

$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots +(-1{)}^{n-1}\frac{x^{2n-1}}{(2n-1)!}+o(x^{2n})$$$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots +(-1{)}^n\frac{x^{2n}}{(2n)!}+o(x^{2n+1})$$$$\tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+o(x^9)$$$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots +\frac{x^n}{n!}+o(x^n)$$$$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots +(-1{)}^{n+1}\frac{x^n}{n}+o(x^n)$$$$(1+x{)}^{\alpha }=1+\alpha x+\frac{\alpha (\alpha -1)}{2!}{x}^2+\cdots +\frac{\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}{x}^n+o(x^n)$$