- $e^x=1+x+\cfrac{x^2}{2!}+\cfrac{x^3}{3!}+\cfrac{x^4}{4!}+...+\cfrac{x^n}{n!}+o(x^n)$
- $\cos x=1-\cfrac{x^2}{2!}+\cfrac{x^4}{4!}-\cfrac{x^6}{6!}+...+\cfrac{(-1)^n}{(2n)!}x^{2n}+o(x^{2n})$ 偶函数
- $\sin x=x-\cfrac{x^3}{3!}+\cfrac{x^5}{5!}-\cfrac{x^7}{7!}+...+\cfrac{(-1)^{n+1}}{(2n+1)!}x^{2n+1}+o(x^{2n+1})$ 奇
$\arcsin x=x+\cfrac{x^3}{3!}$
- $\cfrac{1}{1-x}=1+x+x^2+x^3+...+x^n+o(x^n)$
- $\cfrac{1}{1+x}=1-x+x^2-x^3+...+(-1)^{n}x^n+o(x^n)$
- $\cfrac{1}{1+x^2}=1-x^2+x^4-x^6+...+(-1)^{n}x^{2n}+o(x^{2n})$
- $\ln (1+x)=x-\cfrac{x^2}{2}+\cfrac{x^3}{3}-\cfrac{x^4}{4}+...+\cfrac{(-1){n-1}}{n}x^n+o(x^n)$
- $\arctan x=x-\cfrac{x^3}{3}+\cfrac{x^5}{5}-\cfrac{x^7}{7}+(-1)^{n+1}\cfrac{x^{2n+1}}{2n+1}+o(x^{2n+1})$
- $\tan x=x+\cfrac{x^3}{3}+\cfrac{2}{15}x^5$ 记前两项即可
- $(1+\alpha)^\alpha=1+\alpha x+\frac{\alpha (\alpha-1)}{2!}x^2+\frac{\alpha(\alpha-1) (\alpha-2)}{3!}x^3+\frac{\alpha(\alpha-1)(\alpha-2)...(\alpha-n+1)}{n!}x^n$
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