Title: Mathématiques-PTSI / DL usuels

Q: [latex] $DL_5(0)$ de $\exp$[/latex]
A: [latex] $1+x+\dfrac{x^2}{2} +\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+\dfrac{x^5}{5!}+o(x^5)$[/latex]

Q: [latex] $DL_5(0)$ de $\sin$ [/latex]
A: [latex] $x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}+o(x^6)$[/latex]

Q: [latex] $DL_5(0)$ de $\cos$ [/latex]
A: [latex] $1-\dfrac{x^2}{2}+\dfrac{x^4}{4!}+o(x^5)$ [/latex]

Q: [latex] $DL_5(0)$ de $\tan$[/latex]
A: [latex] $x+\dfrac{x^3}{3}+\dfrac{2x^5}{15}+o(x^5)$ [/latex]

Q: [latex] $DL_5(0)$ de $\Arctan$ [/latex]
A: [latex] $x -\dfrac{x^3}{3}+\dfrac{x^5}{5}+o(x^5)$ [/latex]

Q: [latex] $DL_5(0)$ de $\sh$ [/latex]
A: [latex] $x +\dfrac{x^3}{3!}+\dfrac{x^5}{5!}+o(x^6)$ [/latex]

Q: [latex] $DL_5(0)$ de $\ch$ [/latex]
A: [latex] $1 +\dfrac{x^2}{2}+\dfrac{x^4}{4!}+o(x^5)$[/latex]

Q: [latex] $DL_5(0)$ de $x\mapsto \ln(1+x)$[/latex]
A: [latex] $x -\dfrac{x^2}{2}+\dfrac{x^3}{3}-\dfrac{x^4}{4}+\dfrac{x^5}{5}+o(x^5)$ [/latex]

Q: [latex] $DL_5(0)$ de $x\mapsto \ln(1-x)$[/latex]
A: [latex] $-x -\dfrac{x^2}{2}-\dfrac{x^3}{3}-\dfrac{x^4}{4}-\dfrac{x^5}{5}+o(x^5)$ [/latex]

Q: [latex] $DL_3(0)$ de $x\mapsto (1+x)^{\alpha}$[/latex]
A: [latex] $1+\alpha x+\dfrac{\alpha(\alpha-1)}{2}x^2+\dfrac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3+o(x^3)$ [/latex]

Q: [latex] $DL_5(0)$ de $x\mapsto \dfrac{1}{1+x}$ [/latex]
A: [latex] $1-x+x^2-x^3+x^4-x^5+o(x^5)$ [/latex]

Q: [latex] $DL_5(0)$ de $x\mapsto \dfrac{1}{1-x}$ [/latex]
A: [latex] $1+x+x^2+x^3+x^4+x^5+o(x^5)$ [/latex]