En appliquant la formule de Taylor-Young ou en intégrant ou dérivant des \( DL\) nous obtenons les formules suivantes.
- \( \bullet\)
- \( \dpl{e^x=1+x+\dfrac{x^2}{2}}+\dfrac{x^3}{3!}+\dots+\dfrac{x^n}{n!}+x^n\varepsilon(x)\) .
- \( \bullet\)
- \( \dpl{cos(x)=1-\dfrac{x^2}{2}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+\dots+(-1)^k\dfrac{x^{2k}}{(2k)!}+x^{2k}\varepsilon(x)}\) .
- \( \bullet\)
- \( \dpl{sin(x)=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}+\dots+(-1)^k\dfrac{x^{2k+1}}{(2k+1)!}+x^{2k+1}\varepsilon(x)}\) .
- \( \bullet\)
- \( \dpl{(1+x)^\alpha=1+\alpha x+\alpha(\alpha-1)\dfrac{x^2}{2}+\alpha(\alpha-1)(\alpha-2)\dfrac{x^3}{3!}+\dots+\alpha(\alpha-1)\cdots(\alpha-(n-1))\dfrac{x^n}{n!}+x^n\varepsilon(x)}\)
- \( \alpha = -1\) :
- \( \dfrac{1}{1+x}=1-x+x^2-x^3+\dots+(-1)^nx^n+x^n\varepsilon(x)\)
- \( \alpha = -1\) + \( (\int)\) :
- \( ln(1+x)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\dfrac{x^4}{4}+\dots+(-1)^{n+1}\dfrac{x^n}{n}+x^n\varepsilon(x)\)
- \( \alpha = -1\) + \( (X=-x)\) :
- \( \dfrac{1}{1-X}=1+X+X^2+X^3+X^4+\dots+X^n+X^n\varepsilon(X)\)
- \( \alpha = \dfrac{1}{2}\) :
- \( \sqrt{1+x}=1+\dfrac{1}{2}x+\dfrac{1}{2}\left(-\dfrac{1}{2}\right)\dfrac{x^2}{2!}+\dfrac{1}{2}\left(-\dfrac{1}{2}\right)\left(-\dfrac{3}{2}\right)\dfrac{x^3}{3!}+\dots+\left(\dpl{\prod_{k=1}^{n}}\dfrac{1-2k}{2}\right)\dfrac{x^n}{n!}+x^n\varepsilon(x)\)
