Title: Développements limités usuels en 0 Math1A
Tags: DL exponentielle, log

Q: [latex]$\exp(x)$[/latex]
A: [latex]$1+x+\frac{x^2}{2!}+\frac{x^3}{3!} +\frac{x^4}{4!}+...+\frac{x^n}{n!}+o(x^n)$[/latex]

Q: [latex]$\ln(1+x)$[/latex]
A: [latex]$x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+...+(-1)^{n+1}\frac{x^n}{n}+o(x^n)$[/latex]

Q: [latex]$\frac{1}{1+x}$[/latex]
A: [latex]$1-x+x^{2}-x^{3}+x^{4}+...+(-1)^n x^n+o(x^n)$[/latex]

Q: [latex]$\frac{1}{1-x}$[/latex]
A: [latex]$1+x+x^{2}+x^{3}+x^{4}+...+x^n+o(x^n)$[/latex]

Q: [latex]$\cos(x)$[/latex]
A: [latex]$1-\frac{x^{2}}{2}+\frac{x^{4}}{24}+...+\frac{(-1)^n x^{2n}}{(2n)!}+o(x^{2n+1})$[/latex]

Q: [latex]$\sin(x)$[/latex]
A: [latex]$x-\frac{x^{3}}{6}+\frac{x^{5}}{120} + ...+\frac{(-1)^{n} x^{2n+1}}{(2n+1)!}+o(x^{2n+2})$[/latex]

Q: [latex]$\arctan(x)$[/latex]
A: [latex]$x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+...+\frac{(-1)^n x^{2n+1}}{2n+1}+o(x^{2n+2})$[/latex]

Q: [$]\cosh(x)[/$]
A: [$]1+\frac{x^{2}}{2}+\frac{x^{4}}{24}+\frac{x^{6}}{720} + ... + \frac{x^{2n}}{(2n)!}+o(x^{2n+1})[/$]

Q: [$]\sinh(x)[/$]
A: [$]x+\frac{x^{3}}{6}+\frac{x^{5}}{120}+ ... + \frac{x^{2n+1}}{(2n+1)!}+o(x^{2n+2})[/$]

Q: [$]\text{artanh} (x)[/$]
A: [$]x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\frac{x^{7}}{7}+...+\frac{x^{2n+1}}{2n+1}+o(x^{2n+2})[/$]

Q: [$](1+x)^a[/$]
A: [$]1+ax+\frac{a(a-1)}{2!} x^2 + \frac{a(a-1)(a-2)}{3!} x^3 + ... + \frac{a(a-1)(a-2)...(a-n+1)}{n!} x^n + o(x^n)[/$]

Q: [$]\sqrt{1+x}[/$]
A: [$]1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\frac{5}{128} x^{4} + ... + \frac{1}{2}\times \left(\frac{1}{2} -1\right) \times ...\times \left(\frac{1}{2} -n+1\right)\frac{x^n}{n!} + o(x^n)[/$]