Title: Calcul / DL
Tags: DL

Q: DL de \(\exp(x)\) en 0
A: \(e^x=\displaystyle \sum_{k=0}^n\frac{x^k}{k!}+o(x^n)=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}+\dots\)

Q: DL de \(\ln(1+x)\) en 0
A: \(\ln(1+x)=\displaystyle \sum_{k=1}^n(-1)^k\frac{x^k}{k}+o(x^n)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}+\dots\)

Q: DL de \(\cos(x)\) en 0
A: \(\cos(x)=\displaystyle \sum_{k=0}^n(-1)^k\frac{x^{2k}}{(2k)!}+o(x^{2n})=1-\frac{x^2}{2}+\frac{x^4}{24}+\dots\)

Q: DL de \(\sin(x)\) en 0
A: \(\sin(x)=\displaystyle \sum_{k=0}^n(-1)^k\frac{x^{2k+1}}{(2k+1)!}+o(x^{2n+1})=x-\frac{x^3}{6}+\frac{x^5}{120}+\dots\)

Q: DL de \(\cosh(x)\) en 0
A: \(\cosh(x)=\displaystyle \sum_{k=0}^n\frac{x^{2k}}{(2k)!}+o(x^{2n})=1+\frac{x^2}{2}+\frac{x^4}{24}+\dots\)

Q: DL de \(\sinh(x)\) en 0
A: \(\sinh(x)=\displaystyle \sum_{k=0}^n\frac{x^{2k+1}}{(2k+1)!}+o(x^{2n+1})=x+\frac{x^3}{6}+\frac{x^5}{120}+\dots\)

Q: DL de \(\frac{1}{1-x}\) en 0
A: \(\displaystyle\frac{1}{1-x}= \sum_{k=0}^n x^k+o(x^n)=1+x+x^2+x^3+x^4+x^5+\dots\)

Q: DL de \((1+x)^{\alpha}\) en 0
A: \((1+x)^{\alpha}=\displaystyle \sum_{k=0}^n\binom{\alpha}{k}x^k+o(x^n)\)

Q: DL de \(\arctan(x)\) en 0
A: \(\arctan(x)=\displaystyle \sum_{k=0}^n(-1)^k\frac{x^{2k+1}}{2k+1}+o(x^{2n+1})=x-\frac{x^3}{3}+\frac{x^5}{5}+\dots\)

Q: DL de \(\sqrt{1+x}\) en 0
A: \(\displaystyle\sqrt{1+x}=1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}+\dots\)

Q: DL de \(\frac{1}{\sqrt{1+x}}\) en 0
A: \(\displaystyle\frac{1}{\sqrt{1+x}}=1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+\dots\)

Q: DL de \(\tan(x)\) en 0
A: \(\displaystyle\tan(x)=x+\frac{x^3}{3}+\frac{2x^5}{15}+\dots\)

Q: DL de \(\tanh(x)\) en 0
A: \(\displaystyle\tanh(x)=x-\frac{x^3}{3}+\frac{2x^5}{15}+\dots\)

Q: Formule de Stirling
A: \(n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n\)