常见泰勒展开式

$$\frac{1}{1-x} = 1 + x^2 + x^3 + \cdots + x^n + \cdots = \sum_{n=0} ^\infty x^n$$ $$\frac{1}{1+x} = 1 - x^2 + x^3 - \cdots + (-1)^n x^n + \cdots = \sum_{n=0} ^\infty (-1)^n x^n$$ $$e^x = 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} + \cdots = \sum _{n=0} ^\infty \frac{x^n}{n!}$$ $$sinx = x - \frac{x^3}{3!} + \cdots + \frac{(-1)^n x^{2n+1}}{(2n+1)!} + \cdots = \sum _{n=0} ^{\infty} \frac{(-1)^n x^{2n+1}} {(2n+1)!}$$

常用的带皮亚诺余项的泰勒展开式：

$$e^x = 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} + o(x^n)$$ $$sinx = x - \frac{x^3}{3!} + \cdots + (-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!} + o(x^{2n-1})$$ $$cosx = 1 - \frac{x^2}{2!} + \cdots + (-1)^{n}\frac{x^{2n}}{(2n)!} + o(x^{2n})$$ $$ln(1+x) = x - \frac{x^2}{2} + \cdots + (-1)^{n-1} \frac{x^n}{n} + o(x^n)$$ $$(1+x)^a = 1 + ax + \frac{a(a-1)}{2!}x^2 + \cdots + \frac{a(a-1)\cdots(a-n+1)}{n!} + o(x^n)$$