$$\newcommand{\mtn}{\mathbb{N}}\newcommand{\mtns}{\mathbb{N}^*}\newcommand{\mtz}{\mathbb{Z}}\newcommand{\mtr}{\mathbb{R}}\newcommand{\mtk}{\mathbb{K}}\newcommand{\mtq}{\mathbb{Q}}\newcommand{\mtc}{\mathbb{C}}\newcommand{\mch}{\mathcal{H}}\newcommand{\mcp}{\mathcal{P}}\newcommand{\mcb}{\mathcal{B}}\newcommand{\mcl}{\mathcal{L}} \newcommand{\mcm}{\mathcal{M}}\newcommand{\mcc}{\mathcal{C}} \newcommand{\mcmn}{\mathcal{M}}\newcommand{\mcmnr}{\mathcal{M}_n(\mtr)} \newcommand{\mcmnk}{\mathcal{M}_n(\mtk)}\newcommand{\mcsn}{\mathcal{S}_n} \newcommand{\mcs}{\mathcal{S}}\newcommand{\mcd}{\mathcal{D}} \newcommand{\mcsns}{\mathcal{S}_n^{++}}\newcommand{\glnk}{GL_n(\mtk)} \newcommand{\mnr}{\mathcal{M}_n(\mtr)}\DeclareMathOperator{\ch}{ch} \DeclareMathOperator{\sh}{sh}\DeclareMathOperator{\th}{th} \DeclareMathOperator{\vect}{vect}\DeclareMathOperator{\card}{card} \DeclareMathOperator{\comat}{comat}\DeclareMathOperator{\imv}{Im} \DeclareMathOperator{\rang}{rg}\DeclareMathOperator{\Fr}{Fr} \DeclareMathOperator{\diam}{diam}\DeclareMathOperator{\supp}{supp} \newcommand{\veps}{\varepsilon}\newcommand{\mcu}{\mathcal{U}} \newcommand{\mcun}{\mcu_n}\newcommand{\dis}{\displaystyle} \newcommand{\croouv}{[\![}\newcommand{\crofer}{]\!]} \newcommand{\rab}{\mathcal{R}(a,b)}\newcommand{\pss}[2]{\langle #1,#2\rangle} $$
Bibm@th

Formulaire - Développements en séries entières

Obtenus à partir de $e^x$
$$\begin{array}{rcll} e^x&=&\displaystyle \sum_{n\geq 0}\frac{x^n}{n!},& R=+\infty\\ \cos x&=&\displaystyle \sum_{n\geq 0}\frac{(-1)^n x^{2n}}{(2n)!},& R=+\infty\\ \sin x&=&\displaystyle \sum_{n\geq 0}\frac{(-1)^n x^{2n+1}}{(2n+1)!},& R=+\infty\\ \cosh x&=&\displaystyle \sum_{n\geq 0}\frac{x^{2n}}{(2n)!},& R=+\infty\\ \sinh x&=&\displaystyle \sum_{n\geq 0}\frac{ x^{2n+1}}{(2n+1)!},& R=+\infty\\ \end{array} $$
$1/(1-x),$ et ceux que l'on obtient par intégration à partir de lui
$$\begin{array}{rcll} \displaystyle \frac{1}{1-x}&=&\displaystyle \sum_{n\geq 0}x^n,& R=1\\ \displaystyle \frac{1}{1+x}&=&\displaystyle \sum_{n\geq 0}(-1)^n x^n,& R=1\\ \ln(1+x)&=&\displaystyle \sum_{n\geq 1}\frac{(-1)^{n+1}}{n}x^n,& R=1\\ \ln(1-x)&=&\displaystyle -\sum_{n\geq 1}\frac{x^n}{n},& R=1\\ \arctan(x)&=&\displaystyle \sum_{n\geq 0}\frac{(-1)^n}{2n+1}x^{2n+1}& R=1\\ \textrm{arctanh}(x)&=&\displaystyle \sum_{n\geq 0}\frac{x^{2n+1}}{2n+1}& R=1\\ \end{array} $$
Les autres!
$$\begin{array}{rcll} \displaystyle (1+x)^\alpha&=&\displaystyle 1+\sum_{n\geq 1}\frac{\alpha(\alpha-1)\cdots (\alpha-n+1)}{n!}x^n,& R=1\\ \arcsin(x)&=&\displaystyle \sum_{n\geq 0}\frac{(2n)!}{2^{2n}(n!)^2 (2n+1)}x^{2n+1},&R=1\\ \textrm{arcsinh}(x)&=&\displaystyle \sum_{n\geq 0}\frac{(-1)^n(2n)!}{2^{2n}(n!)^2 (2n+1)}x^{2n+1},&R=1\\ \end{array}$$