Fonctions d'une variable réelle/Fiche/Développements limités

Fiche mémoire sur les développements limités usuels

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