泰勒公式
三角函数
\[\sin x = x - \frac{x^2}{3!} + \frac{x^5}{5!} + (-1)^{2n-1}\frac{x^{2n-1}}{(2n-1)!} + O(x^{2n-1}) \]
\[\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + (-1)^{2n-1}\frac{x^{2n}}{(2n)!} + O(x^{2n}) \]
\[\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + O(x^5) \]
\[\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} + O(x^5) \]
\[\arcsin x = x + \frac{x^3}{3!} + O(x^3) \]
\[\arccos x = \frac{\pi}{2} - x - \frac{x^3}{3} + O(x^3) \]
常用函数
\[e^x = 1 + x + \frac{x^2}{2!} + ...+\frac{x^n}{n!} +O(x^n) \]
\[\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} +O(x^3) \]
\[(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha - 1)x^2}{2!}+...+\frac{\alpha...(\alpha-n+1)x^n}{n!}+O(x^n) \]