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\newcommand{\Arctan}{\mathrm{Arctan}}

\begin{document}
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\title{Développements Limités}
\author{Aubin SIONVILLE}
\date{MPI Clemenceau - 2021-2023}
\maketitle

\centering

\[
\begin{array}{c|c}
    \hline \\
    \frac{1}{1-x} & 1 + x + x^2 + x^3 + \dots + \mathrm{o}(x^n) \\ \\ \hline \\
    \frac{1}{1+x} & 1 - x + x^2 - x^3 + \dots + \mathrm{o}(x^n) \\ \\ \hline \\
    \ln(1+x) & x - \frac{x^2}{2} + \frac{x^3}{3} + \dots + \mathrm{o}(x^n) \\ \\ \hline \\
    \Arctan(x) & x - \frac{x^3}{3} + \frac{x^5}{5} + \dots + \mathrm{o}(x^{2n+1}) \\ \\ \hline \\
    \mathrm{e}^x & 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \mathrm{o}(x^n) \\ \\ \hline \\
    \sin(x) & x - \frac{x^3}{3!} + \frac{x^5}{5!} + \dots + \mathrm{o}(x^{2n+1}) \\ \\ \hline \\
    \cos(x) & 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \dots + \mathrm{o}(x^{2n}) \\ \\ \hline \\
    (1+x)^\alpha & 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!} x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!} x^3 + \dots + \mathrm{o}(x^n) \\ \\ \hline \\
    \sqrt{1+x} & 1 - \frac{x}{2} + \frac{x^2}{8} - \frac{x^3}{16} + \dots + \mathrm{o}(x^n) \\ \\ \hline \\
    \tan(x) & x + \frac{x^3}{3} + \frac{2x^5}{15} + \dots + \mathrm{o}(x^5) \\ \\ \hline
\end{array}\]

\end{document}
