泰勒展开
$f(x) 在 x = x_0 的展开$ \(f(x) = \sum_{n=0}^{\infty} \dfrac{f^{(n)}(x_0)}{n!}(x-x_0)^n\)
皮亚诺余项
\[f(x) = \sum_{k=0}^{n} \dfrac{f^{(k)}(x_0)}{k!}(x-x_0)^k + o((x-x_0)^{n})\]拉格朗日余项
\[f(x) = \sum_{k=0}^{n} \dfrac{f^{(k)}(x_0)}{k!}(x-x_0)^{k+1} + \dfrac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}\] \[\xi \in (x_0, x)\]# 幂级数的展开
\[\begin{array}{rlll} e^x &= \sum_{n=0}^{\infty} \dfrac{x^n}{n!} & \dfrac{x^n}{n!} \\\\ \cos(x) &= \sum_{n=0}^{\infty} (-1)^n\dfrac{x^{2n}}{({2n})!} & \dfrac{x^n}{n!} \quad \text{交错偶项}\\\\ \sin(x) &= \sum_{n=0}^{\infty} (-1)^n\dfrac{x^{2n+1}}{({2n+1})!} & \dfrac{x^n}{n!} \quad \text{交错奇项} \\\\ \operatorname{ch}(x) &= \sum_{n=0}^{\infty} \dfrac{x^{2n}}{2n!} & \dfrac{x^n}{n!} \quad \text{偶项} \\\\ \operatorname{sh}(x) &= \sum_{n=0}^{\infty} \dfrac{x^{2n+1}}{2n+1!} & \dfrac{x^n}{n!} \quad \text{奇项}\\\\ \dfrac{1}{1-x} &= \sum_{n=0}^{\infty} x^n & x^n\\\\ \dfrac{1}{1+x} &= \sum_{n=0}^{\infty} (-1)^n x^n & x^n \quad \text{交错}\\\\ \ln(1+x) &= \sum_{n=1}^{\infty} (-1)^{n+1}\dfrac{x^{n}}{n} & \dfrac{x^n}{n} \quad \text{交错}\\\\ \arctan(x) &= \sum_{n=0}^{\infty} (-1)^n\dfrac{x^{2n+1}}{2n+1} & \dfrac{x^n}{n} \quad \text{交错奇项}\\\\ \operatorname{arctanh}(x) &= \sum_{n=0}^{\infty}\dfrac{x^{2n+1}}{2n+1} & \dfrac{x^n}{n} \quad \text{奇项}\\\\ (1+x)^\alpha &= 1 + \sum_{n=1}^{\infty} \dfrac{\alpha(\alpha-1\cdots(\alpha-n+1))}{n!}x^n &= \sum_{n=1}^{\alpha} C^\alpha_n x^n \end{array}\]总结，出现奇项偶项的为三角函数的展开，出现交错的首项均为正项
特别的
\[\pi = 4\sum_{n=0}^{\infty} \dfrac{(-1)^n}{2n+1}\]