Développements limités usuels au voisinage de 0

\begin{align*} e^x &\ \ =\ \ 1 + \frac{x}{1!}+\frac{x^2}{2!}+ \cdots + \frac{x^n}{n!}\ \ + \ \ o(x^n) = \sum_{k=0}^n \frac{x^k}{k!} \ \ + \ \ o(x^n) \\[1.2em] \text{cos } x &\ \ =\ \ 1 - \frac{x^2}{2!}+\frac{x^4}{4!}- \cdots + (-1)^n \cdot \frac{x^{2n}}{(2n)!}\ \ + \ \ o(x^{2n+1}) = \sum_{k=0}^n (-1)^{k} \frac{x^{2k}}{(2k)!} \ \ + \ \ o(x^{2n+1}) \\[0.5em] \text{sin } x &\ \ =\ \ x - \frac{x^3}{3!}+\frac{x^5}{5!}- \cdots + (-1)^n \cdot \frac{x^{2n+1}}{(2n+1)!}\ \ + \ \ o(x^{2n+2}) = \sum_{k=0}^n (-1)^{k} \frac{x^{2k+1}}{(2k+1)!} \ \ + \ \ o(x^{2n+2}) \\[0.5em] \text{tan } x &\ \ =\ \ x + \frac{x^3}{3} + \frac{2}{15}x^5 + \frac{17}{315}x^7\ \ + \ \ o(x^{8}) \\[1.2em] \text{ch } x &\ \ =\ \ 1 + \frac{x^2}{2!}+\frac{x^4}{4!}+ \cdots + \frac{x^{2n}}{(2n)!}\ \ + \ \ o(x^{2n+1}) = \sum_{k=0}^n \frac{x^{2k}}{(2k)!} \ \ + \ \ o(x^{2n+1}) \\[0.5em] \text{sh } x &\ \ =\ \ x + \frac{x^3}{3!}+\frac{x^5}{5!}+ \cdots + \frac{x^{2n+1}}{(2n+1)!}\ \ + \ \ o(x^{2n+2}) = \sum_{k=0}^n \frac{x^{2k+1}}{(2k+1)!} \ \ + \ \ o(x^{2n+2}) \\[0.5em] \text{th } x &\ \ =\ \ x - \frac{x^3}{3} + \frac{2}{15}x^5 - \frac{17}{315}x^7\ \ + \ \ o(x^{8}) \\[0.5em] \end{align*} \begin{align*} \text{ln}\, (1+x)&\ \ =\ \ x - \frac{x^2}{2}+\frac{x^3}{3}- \cdots + (-1)^{n-1}\cdot\frac{x^n}{n}\ \ + \ \ o(x^n) = \sum_{k=1}^n (-1)^{k+1} \frac{x^k}{k} \ \ + \ \ o(x^n) \\[0.5em] (1+x)^\alpha &\ \ =\ \ 1 + \alpha x + \frac{\alpha(\alpha - 1)}{2!}x^2+ \cdots + \frac {\alpha(\alpha-1)\cdots(\alpha - n+1)}{n!} x^n \ \ + \ \ o(x^n)\\ &\ \ =\ \ \sum_{k=0}^n \binom{\alpha}{k} x^k \ \ + \ \ o(x^n) \\[1.2em] \frac{1}{1+x}&\ \ =\ \ 1 -x+x^2-\cdots+(-1)^n x^n \ \ +\ \ o(x^n) = \sum_{k=0}^n (-1)^k x^k \ \ +\ \ o(x^n) \\[0.5em] \frac{1}{1-x}&\ \ =\ \ 1 + x+x^2+ \cdots+ x^n\ \ +\ \ o(x^n) = \sum_{k=0}^n x^k \ \ +\ \ o(x^n) \\[0.5em] \sqrt{1+x}&\ \ =\ \ 1 + \frac{x}{2} - \frac{1}{8}x^2- \cdots + (-1)^{n-1}\cdot\frac{1\cdot1\cdot3\cdot5\cdots(2n-3)}{2^n n!}x^n\ \ + \ \ o(x^n) \\[0.5em] \frac{1}{\sqrt{1+x}}&\ \ =\ \ 1 - \frac{x}{2} + \frac{3}{8}x^2- \cdots + (-1)^{n}\cdot\frac{1\cdot3\cdot5\cdots(2n-1)}{2^n n!} x^n\ \ + \ \ o(x^n) \\[1.2em] \text{arccos } x &\ \ =\ \ \frac{\pi}{2} - x - \frac{1}{2}\frac{x^3}{3} - \frac{1\cdot3}{2\cdot4}\frac{x^5}{5}- \cdots - \frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)} \frac{x^{2n+1}}{2n+1}\ \ + \ \ o(x^{2n+2}) \\[0.5em] \text{arcsin } x &\ \ =\ \ x + \frac{1}{2}\frac{x^3}{3} + \frac{1\cdot3}{2\cdot4}\frac{x^5}{5}+ \cdots + \frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)} \frac{x^{2n+1}}{2n+1}\ \ + \ \ o(x^{2n+2}) \\[0.5em] \text{arctan } x &\ \ =\ \ x - \frac{x^3}{3}+\frac{x^5}{5}+ \cdots + (-1)^{n}\cdot\frac{x^{2n+1}}{2n+1}\ \ + \ \ o(x^{2n+2}) \end{align*}