Fonctions d'une variable réelle/Fiche/Développements limités

$\begin{array}{cccc}& & \text{A~l'ordre~}n& \text{Premiers~termes}\\ \cos (x)& =& \sum _{k=0}^n(-1{)}^k\frac{x^{2k}}{(2k)!}+\underset{x\to 0}{\mathrm{o}}(x^{2n+1})& \cos (x)=1-\frac{x^2}{2}+\frac{x^4}{24}+\underset{x\to 0}{\mathrm{o}}(x^5)\\ \sin (x)& =& \sum _{k=0}^n(-1{)}^k\frac{x^{2k+1}}{(2k+1)!}+\underset{x\to 0}{\mathrm{o}}(x^{2n+2})& \sin (x)=x-\frac{x^3}{6}+\frac{x^5}{120}+\underset{x\to 0}{\mathrm{o}}(x^6)\\ \mathrm{c}\mathrm{h}(x)& =& \sum _{k=0}^n\frac{x^{2k}}{(2k)!}+\underset{x\to 0}{\mathrm{o}}(x^{2n+1})& \mathrm{c}\mathrm{h}(x)=1+\frac{x^2}{2}+\frac{x^4}{24}+\underset{x\to 0}{\mathrm{o}}(x^5)\\ \mathrm{s}\mathrm{h}(x)& =& \sum _{k=0}^n\frac{x^{2k+1}}{(2k+1)!}+\underset{x\to 0}{\mathrm{o}}(x^{2n+2})& \mathrm{s}\mathrm{h}(x)=x+\frac{x^3}{6}+\frac{x^5}{120}+\underset{x\to 0}{\mathrm{o}}(x^6)\\ e^x& =& \sum _{k=0}^n\frac{x^k}{k!}+\underset{x\to 0}{\mathrm{o}}(x^n)& e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\underset{x\to 0}{\mathrm{o}}(x^3)\\ \ln (1+x)& =& \sum _{k=1}^n(-1{)}^{k+1}\frac{x^k}{k}+\underset{x\to 0}{\mathrm{o}}(x^n)& \ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\underset{x\to 0}{\mathrm{o}}(x^3)\\ \frac{1}{1+x}& =& \sum _{k=0}^n(-1{)}^k{x}^k+\underset{x\to 0}{\mathrm{o}}(x^n)& \frac{1}{1+x}=1-x+x^2-x^3+\underset{x\to 0}{\mathrm{o}}(x^3)\\ \frac{1}{1-x}& =& \sum _{k=0}^n{x}^k+\underset{x\to 0}{\mathrm{o}}(x^n)& \frac{1}{1-x}=1+x+x^2+x^3+\underset{x\to 0}{\mathrm{o}}(x^3)\end{array}$