常见泰勒展开式

\[f(x)=\sum_{k=0}^{n}\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k +o\left((x-x_0)^n\right) \]

\[f(x)=\sum_{k=0}^{n}\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k +\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1} \]

\[e^x =1+x+\frac{x^2}{2!}+\frac{x^3}{3!} +\cdots+\frac{x^n}{n!}+o(x^n) \]

\[e^{-x} =1-x+\frac{x^2}{2!}-\frac{x^3}{3!} +\cdots+\frac{(-1)^n x^n}{n!}+o(x^n) \]

\[\sin x =x-\frac{x^3}{3!}+\frac{x^5}{5!} -\frac{x^7}{7!}+\cdots \]

\[\cos x =1-\frac{x^2}{2!}+\frac{x^4}{4!} -\frac{x^6}{6!}+\cdots \]

\[\tan x =x+\frac{x^3}{3}+\frac{2x^5}{15} +\frac{17x^7}{315}+o(x^7) \]

\[\arcsin x =x+\frac{x^3}{6}+\frac{3x^5}{40} +\frac{5x^7}{112}+o(x^7) \]

\[\arctan x =x-\frac{x^3}{3}+\frac{x^5}{5} -\frac{x^7}{7}+\cdots \]

\[\ln(1+x) =x-\frac{x^2}{2}+\frac{x^3}{3} -\frac{x^4}{4}+\cdots +\frac{(-1)^{n-1}x^n}{n}+o(x^n) \]

\[\ln(1-x) =-x-\frac{x^2}{2}-\frac{x^3}{3} -\frac{x^4}{4}-\cdots-\frac{x^n}{n}+o(x^n) \]

\[(1+x)^\alpha =1+\alpha x +\frac{\alpha(\alpha-1)}{2!}x^2 +\frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3 +\cdots \]

\[\sqrt{1+x} =1+\frac{x}{2}-\frac{x^2}{8} +\frac{x^3}{16}-\frac{5x^4}{128}+o(x^4) \]

\[\frac{1}{\sqrt{1+x}} =1-\frac{x}{2}+\frac{3x^2}{8} -\frac{5x^3}{16}+\frac{35x^4}{128}+o(x^4) \]

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

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

\[x-\sin x =\frac{x^3}{6}-\frac{x^5}{120}+o(x^5) \]

\[1-\cos x =\frac{x^2}{2}-\frac{x^4}{24} +\frac{x^6}{720}+o(x^6) \]

\[\tan x-x =\frac{x^3}{3}+\frac{2x^5}{15}+o(x^5) \]

\[\tan x-\sin x =\frac{x^3}{2}+\frac{x^5}{8}+o(x^5) \]

\[\arcsin x-\arctan x =\frac{x^3}{2}+\frac{x^5}{8}+o(x^5) \]

\[e^x-1-x =\frac{x^2}{2}+\frac{x^3}{6}+o(x^3) \]

\[e^x-1-x-\frac{x^2}{2} =\frac{x^3}{6}+o(x^3) \]

\[x-\ln(1+x) =\frac{x^2}{2}-\frac{x^3}{3} +\frac{x^4}{4}+o(x^4) \]

\[\ln(1+x)-x+\frac{x^2}{2} =\frac{x^3}{3}+o(x^3) \]

\[\frac{\sin x}{x} =1-\frac{x^2}{6}+\frac{x^4}{120}+o(x^4) \]

\[\frac{\tan x}{x} =1+\frac{x^2}{3}+\frac{2x^4}{15}+o(x^4) \]

\[\ln(\cos x) =-\frac{x^2}{2}-\frac{x^4}{12}+o(x^4) \]

\[\ln\frac{\sin x}{x} =-\frac{x^2}{6}-\frac{x^4}{180}+o(x^4) \]

\[(1+x)^{1/x} =e\left(1-\frac{x}{2}+\frac{11x^2}{24}+o(x^2)\right) \]