Les développements limités usuels en 0

 \begin{array}{rcl}
e^x & = & \displaystyle \sum_{k=0}^n \dfrac{x^k}{k!}+ o\left( x^n\right)\\
\cos(x) & = & \displaystyle \sum_{k=0}^n (-1)^k\dfrac{x^{2k}}{(2k)!}+ o\left( x^{2n}\right)(\text{ou } o(x^{2n+1}))\\
\sin(x) & = & \displaystyle \sum_{k=0}^n (-1)^k\dfrac{x^{2k+1}}{(2k+1)!}+ o\left( x^{2n+1}\right)(\text{ou } o(x^{2n+2}))\\
\text{ch}(x) & = & \displaystyle \sum_{k=0}^n \dfrac{x^{2k}}{(2k)!}+ o\left( x^{2n}\right)(\text{ ou } o(x^{2n+1}))\\
\text{sh}(x) & = & \displaystyle \sum_{k=0}^n \dfrac{x^{2k+1}}{(2k+1)!}+ o\left( x^{2n+1}\right)(\text{ou } o(x^{2n+2}))\\
   \end{array}

 \begin{array}{rcl}
\alpha \in \mathbb{R},(1+x)^\alpha & = & 1 + \alpha x + \dfrac{\alpha(\alpha-1)}{2}x^2+ \ldots+ \\
&&\dfrac{\alpha(\alpha-1)\ldots (\alpha-(n-1))}{n!}x^n+o(x^n)\\
\dfrac{1}{1-x} & = & \displaystyle \sum_{k=0}^n x^k+ o\left( x^{n}\right)\\
\dfrac{1}{1+x} & = & \displaystyle \sum_{k=0}^n (-1)^kx^k+ o\left( x^{n}\right)\\
\dfrac{1}{(1-x)^2} & = & \displaystyle \sum_{k=0}^n (k+1)x^k+ o\left( x^{n}\right)\\
\sqrt{1+x} & = & \displaystyle 1+\dfrac{x}{2}- \dfrac{x^2}{8}+\ldots+\\
&& (-1)^{n-1}\dfrac{1\times 3\times \ldots \times (2n-3)}{2\times 4 \times \ldots \times (2n) } x^n+o\left( x^{n}\right)\\
\end{array}

 \begin{array}{rcl}
\ln (1-x) & = & \displaystyle -\sum_{k=1}^n \dfrac{x^k}{k}+ o\left( x^{n}\right)\\
\ln (1+x) & = & \displaystyle \sum_{k=1}^n (-1)^{k-1}\dfrac{x^k}{k}+ o\left( x^{n}\right)\\
\end{array}

 \begin{array}{rcl}
\tan x & = & \displaystyle x+\dfrac{1}{3}x^3+\dfrac{2}{15}x^5+ \dfrac{17}{315}x^7+ o\left( x^{8}\right)\\ \\
\th x & = & \displaystyle x-\dfrac{1}{3}x^3+\dfrac{2}{15}x^5- \dfrac{17}{315}x^7+ o\left( x^{8}\right)\\\end{array}

 \begin{array}{rcl}
\arccos x & = & \displaystyle \dfrac{\pi}{2}-(x+\dfrac{x^3}{6}+\dfrac{3}{8}\dfrac{x^5}{5}+\ldots + \\
 &&\dfrac{1 \times 3 \times \ldots \times(2n-1)}{2 \times 2 \times \ldots \times(2n)}\dfrac{x^{2n+1}}{2n+1} )+ o\left( x^{2n+2}\right)\\
\arcsin x & = & \displaystyle x+\dfrac{x^3}{6}+\dfrac{3}{8}\dfrac{x^5}{5}+\ldots + \\
 &&\dfrac{1 \times 3 \times \ldots \times(2n-1)}{2 \times 2 \times \ldots \times(2n)}\dfrac{x^{2n+1}}{2n+1})+ o\left( x^{2n+2}\right)\\
\arctan x & = &\displaystyle  \sum_{k=0}^n(-1)^k \dfrac{x^{2k+1}}{2k+1}+ o\left( x^{2n+2}\right)\\
\text{argch } \left( \dfrac{1+x}{\sqrt{x}}\right) & = & \sqrt{2} - \dfrac{\sqrt 2}{\sqrt{12}}x +\dfrac{3\sqrt{2}}{160}x^2 +\dfrac{5\sqrt{2}}{896}x^3+\dfrac{35\sqrt{2}}{18432}x^4+o(x^4)\\
\text{argsh } x & = &  \displaystyle x-\dfrac{x^3}{6}+\dfrac{3}{8}\dfrac{x^5}{5}+\ldots + \\
 &&(-1)^n\dfrac{1 \times 3 \times \ldots \times(2n-1)}{2 \times 2 \times \ldots \times(2n)}\dfrac{x^{2n+1}}{2n+1})+ o\left( x^{2n+2}\right)\\
\text{argth } x & = &\displaystyle  \sum_{k=0}^n \dfrac{x^{2k+1}}{2k+1}+ o\left( x^{2n+2}\right)\\
\end{array}