高数极限-麦克劳林展开式

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

\[e^x = 1 + x + \frac{x^2}{2} + ...+\frac{x^n}{n!} +O(x^n) \]

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

\[(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha - 1)x^2}{2!}+...+\frac{\alpha...(\alpha-n+1)x^n}{n!}+O(x^n) \]