# 考研数学常用的泰勒公式（麦克劳林公式）汇总

## 二、正文

\begin{aligned} \begin{rcases} \textcolor{springgreen}{\boldsymbol{ e^{x} }} & = \sum_{\textcolor{orangered}{n=0}}^{\infty} \frac{1}{n !} x^{n} \\ \\ & = 1+x+\frac{1}{2 !} x^{2}+\cdots \end{rcases} & \textcolor{yellow}{ x \in (-\infty, +\infty) } \\ \\ \\ \begin{rcases} \textcolor{springgreen}{\boldsymbol{ \sin x }} & = \sum_{\textcolor{orangered}{n=0}}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1} \\ \\ & = x-\frac{1}{3 !} x^{3}+\frac{1}{5 !} x^{5}+\cdots \end{rcases} & \textcolor{yellow}{ x \in(-\infty,+\infty) } \\ \\ \\ \begin{rcases} \textcolor{springgreen}{\boldsymbol{ \cos x }} & = \sum_{\textcolor{orangered}{n=0}}^{\infty} \frac{(-1)^{n}}{(2 n) !} x^{2 n} \\ \\ & = 1-\frac{1}{2 !} x^{2}+\frac{1}{4 !} x^{4}+\cdots \end{rcases} & \textcolor{yellow}{ x \in(-\infty,+\infty) } \\ \\ \\ \begin{rcases} \textcolor{springgreen}{\boldsymbol{ \ln (1+x) }} & = \sum_{\textcolor{orangered}{n=0}}^{\infty} \frac{(-1)^{n}}{n+1} x^{n+1} \\ \\ & = \sum_{\textcolor{magenta}{n=1}}^{\infty} \frac{(-1)^{n-1}}{n} x^{n} \\ \\ & = x-\frac{1}{2} x^{2}+\frac{1}{3} x^{3}+\cdots \end{rcases} & \textcolor{yellow}{ x \in(-1,1] } \\ \\ \\ \begin{rcases} \textcolor{springgreen}{\boldsymbol{ \frac{1}{1-x} }} & = \sum_{\textcolor{orangered}{n=0}}^{\infty} x^{n}\\ \\ & = 1+x+x^{2}+x^{3}+\cdots \end{rcases} & \textcolor{yellow}{ x \in(-1,1) } \\ \\ \\ \begin{rcases} \textcolor{springgreen}{\boldsymbol{ \frac{1}{1+x} }} & =\sum_{\textcolor{orangered}{n=0}}^{\infty}(-1)^{n} x^{n} \\ & = 1-x+x^{2}-x^{3}+\cdots \end{rcases} & \textcolor{yellow}{ x \in(-1,1) } \\ \\ \\ \begin{rcases} \textcolor{springgreen}{\boldsymbol{ (1+x)^{\alpha} }} & = 1+\sum_{\textcolor{magenta}{n=1}}^{\infty} \frac{\alpha(\alpha-1) \cdots(\alpha-n+1)}{n !} x^{n} \\ \\ & = 1+\alpha x+\frac{\alpha(\alpha-1)}{2 !} x^{2}+\cdots \end{rcases} & \textcolor{yellow}{ x \in(-1,1) } \end{aligned}

⟬1⟭ 当 $\textcolor{yellow}{x}$ $\textcolor{yellow}{\in}$ $\textcolor{yellow}{[-1,1]}$ 时，有：

\begin{aligned} \textcolor{springgreen}{\boldsymbol{\arctan x}} & =\sum_{\textcolor{orangered}{n=0}}^{\infty} \frac{(-1)^{n}}{2 n+1} x^{2 n+1} \\ \\ & = x-\frac{1}{3} x^{3}+\frac{1}{5} x^{5}+\cdots \end{aligned}

⟬2⟭ 当 $\textcolor{yellow}{x}$ $\textcolor{yellow}{\in}$ $\textcolor{yellow}{(-1,1)}$ 时，有：

\begin{aligned} \textcolor{springgreen}{\boldsymbol{\arcsin x}} & = \sum_{\textcolor{orangered}{n=0}}^{\infty} \frac{(2 n) !}{4^{n}(n !)^{2}(2 n+1)} x^{2n+1} \\ \\ & = x+\frac{1}{6} x^{3}+\frac{3}{40} x^{5}+\frac{5}{112} x^{7}+\frac{35}{1152} x^{9}+\cdots \end{aligned}

⟬3⟭ 当 $\textcolor{yellow}{x}$ $\textcolor{yellow}{\in}$ $\textcolor{yellow}{(-\frac{\pi}{2}, \frac{\pi}{2})}$ 时，有：

\begin{aligned} \textcolor{springgreen}{\boldsymbol{\tan x}} & = \sum_{\textcolor{magenta}{n=1}}^{\infty} \frac{\textcolor{#00bffe}{B_{2 n} } (-4)^{n}\left(1-4^{n}\right)}{(2 n) !} x^{2 n-1} \\ \\ & = x+\frac{1}{3} x^{3}+\frac{2}{15} x^{5}+\frac{17}{315} x^{7}+\frac{62}{2835} x^{9}+\frac{1382}{155925} x^{11}+ \\ & \frac{21844}{6081075} x^{13}+\frac{929569}{638512875} x^{15}+\cdots \end{aligned}