Développements limités
- Exponentielle
$$e^{x} = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^n}{n!}+o(x^n)$$
- Cosinus
$$\cos(x) = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots+(-1)^n\frac{x^{2n}}{(2n)!}+o(x^{2n+1})$$
- Sinus
$$\sin(x) = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots+(-1)^n\frac{x^{2n+1}}{(2n+1)!}+o(x^{2n+2})$$
- Cosinus Hyperbolique
$${\rm ch}(x) = 1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots+\frac{x^{2n}}{(2n)!}+o(x^{2n+1})$$
- Sinus Hyperbolique
$${\rm sh}(x) = x+\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots+\frac{x^{2n+1}}{(2n+1)!}+o(x^{2n+2})$$
- Fractions élémentaires
$$\frac1{1-x} = 1+x+x^2+x^3+\cdots+x^n+o(x^n)$$
$$\frac1{1+x} = 1-x+x^2-x^3+\cdots+(-1)^nx^n+o(x^n)$$
- Logarithme Népérien
$$\ln(1-x) = -x-\frac{x^2}2-\frac{x^3}3-\cdots-\frac{x^{n}}n+o(x^{n})$$
$$\ln(1+x) = x-\frac{x^2}2+\frac{x^3}3-\cdots+(-1)^{(n-1)}\frac{x^{n}}n+o(x^{n})$$
- Arc Tangente
$${\rm Arctan}(x)=x-\frac{x^3}3+\frac{x^5}5- \cdots +(-1)^n\frac{x^{2n+1}}{2n+1}+o(x^{2n+2})$$
- Puissance
$$(1+x)^a = 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-1)(a-2)}{3!}x^3 + \cdots + \frac{a(a-1)(a-2)...(a-(n-1))}{n!}x^n + o(x^n)$$
- Racine carrée
$$\sqrt{1 + x} = 1 + \frac1{2} x - \frac1{8} x^2 + \frac1{16} x^3 - o(x^3)$$