常用的泰勒公式

$$\begin{array}{rl}{e}^x& =1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+o(x^3)\\ & \\ \ln (1+x)& =x-\frac{x^2}{2}+\frac{x^3}{3}+o(x^3)\\ & \\ (1+x{)}^{\alpha }& =1+\alpha x+\frac{\alpha (\alpha -1)}{2!}{x}^2+\frac{\alpha (\alpha -1)(\alpha -2)}{3!}{x}^3+o(x^3)\\ & \\ \frac{1}{1-x}& =1+x+x^2+x^3+o(x^3)\\ & \\ \frac{1}{1+x}& =1-x+x^2-x^3+o(x^3)\\ & \\ \sin x& =x-\frac{x^3}{3！}+\frac{x^5}{5!}+o(x^5)\\ & \\ \arcsin x& =x+\frac{1}{2}\times \frac{x^3}{3}+\frac{1\times 3}{2\times 4}\times \frac{x^5}{5}+\frac{1\times 3\times 5}{2\times 4\times 6}\times \frac{x^7}{7}+o(x^7)\\ & \\ \cos x& =1-\frac{x^2}{2!}+\frac{x^4}{4!}+o(x^4)\\ & \\ \sqrt{1+x}& =1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}+o(x^3)\\ & \\ \tan x& =x+\frac{x^3}{3}+\frac{2x^5}{15}+o(x^5)\\ & \\ \arctan x& =x-\frac{x^3}{3}+\frac{x^5}{5}+o(x^5)\end{array}$$