$$\def \N {\mathbb N} \def \R {\mathbb R}$$
$$\left\{ \begin{array}{l} \dot f = - \alpha g f \\ \dot g = \alpha g f - \beta g \\ \dot h = \beta g \end{array}\right.$$
$$2\pi=6.28318530718\cdots$$
$$\sqrt 2=1.41421356237\cdots$$
$$\sum_{n=0}^\infty \frac 1 {n}=+\infty$$
$$1+\frac 1 2 + \frac 1 4 + \frac 1 8 + \cdots = 2$$
$$e^{i\pi}=-1$$
$$\ln(1-x)=-x-\frac{x^2} 2-\frac{x^3}3 - \cdots$$
$$\sin(x)=x-\frac{x^3} 6+\frac{x^5}{120} + \cdots$$
$$\lim_{n\rightarrow \infty}(1+x/n)^n=e^x$$
$$\frac{\pi^2} 6=\sum_n \frac 1 {n^2}$$
$$\lim_{x\rightarrow 0} \frac{f(x)}{g(x)}=\frac{f'(x)}{g'(x)}$$
$$x_{n+1}=x_n-h \nabla J(x_n)$$
$$e^{x+y}=e^xe^y$$
$$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)} h$$
Analyse et méthodes numériques
Développements limités
IUT Nice Sophia-Antipolis

## Dérivation revisitée

$$f(x+\delta x) =f(x)+ \delta x f'(x) + \color{#ffff88}{r(\delta x)}$$
$$\color{#ffff88}{r(\delta x)}=f(x+\delta x) -(f(x)+ \delta x f'(x)).$$
$$\color{#ffff88}{r(\delta x)}=f(x+\delta x) -f(x) - \delta x f'(x)).$$
$$\frac{\color{#ffff88}{r(\delta x)}}{\delta x} = \frac{f(x+\delta x) - f(x)- \delta x f'(x)}{\delta x}$$
$$\frac{\color{#ffff88}{r(\delta x)}}{\delta x} = \frac{f(x+\delta x) - f(x)}{\delta x} - f'(x)$$
$$\frac{\color{#ffff88}{r(\delta x)}}{\delta x} \xrightarrow[\delta x \rightarrow 0]{} 0$$
Notation de Landau
petit "o"
$$\color{#ffff88}{r(x)}=\color{red}{o(x)} \Longleftrightarrow \color{#9999ee}{\lim_{x\rightarrow 0} \frac{r(x)} x =0}$$
$$f(x+\delta x) =f(x)+ \delta x f'(x) + \color{#ffff88}{r(\delta x)}$$
$$f(x+\delta x) =f(x)+ \delta x f'(x) + \color{red}{o(\delta x)}$$
et $$\color{#9999ee}{\lim_{x\rightarrow 0} \frac{r(x)} x =0}$$

## Ordres supérieurs

$$r(x)=o(x^n) \Longleftrightarrow \lim_{x\rightarrow 0} \frac{r(x)} {x^n}=0.$$

## Formule de Taylor

$$f(x+\delta x)=\color{#ee9999}{f(x)+f'(x) \delta x}+ o(\delta x)$$
$$f(x+\delta x)=\color{#ee9999}{f(x)+f'(x)\delta x } \\ \color{#99ee99}{+\frac{f''(x)} 2 \delta x^2} + o(\delta x^2)$$
$$f(x+\delta x)=\color{#ee9999}{f(x)+ f'(x) \delta x } \\ \color{#99ee99}{+\frac{f''(x)} 2 \delta x^2} \\ \color{#9999ee}{+\frac{f^{(3)}(x)}{3!} } \delta x^3+ o(\delta x^3)$$
$$f(x+\delta x)=\color{#ee9999}{f(x)+f'(x) \delta x } \\ \color{#99ee99}{+\frac{ f''(x)} 2 \delta x^2} \\ \color{#9999ee}{+\frac{f^{(3)}(x)}{3!} \delta x^3} \\ \color{#ffff88}{+\frac{ f^{(4)}(x)}{4!} \delta x^4} + o(\delta x^4)$$
$$f(x+\delta x)=\color{#ee9999}{f(x)+f'(x) \delta x } \\ \color{#99ee99}{+\frac{ f''(x)} 2 \delta x^2} \\ \color{#9999ee}{+\frac{f^{(3)}(x)}{3!} \delta x^3} \\ \color{#ffff88}{+\frac{ f^{(4)}(x)}{4!} \delta x^4} \\ +\cdots+\frac{ f^{(n)}(x)} {n!} \delta x^n + o(\delta x^n)$$

## Développement de $\cos(x)$

$$\cos(\delta x)=\color{#ee9999}{\cos(0)+\cos'(0) \delta x } \\ \color{#99ee99}{+\frac{ \cos''(0)} 2 \delta x^2} \\ \color{#9999ee}{+\frac{\cos^{(3)}(0)}{3!} \delta x^3} \\ \color{#ffff88}{+\frac{\cos^{(4)}(0)}{4!} \delta x^4} \\ +\cdots+\frac{ \cos^{(n)}(0)}{n!} \delta x^n + o(\delta x^n)$$
$$\cos(\delta x)=\color{#ee9999}{\cos(0)+\cos'(0) \delta x } \\ \color{#99ee99}{+\frac{ \cos''(0)} 2 \delta x^2} \\ \color{#9999ee}{+\frac{\cos^{(3)}(0)}{3!} \delta x^3} \\ \color{#ffff88}{+\frac{\cos^{(4)}(0)}{4!} \delta x^4} \\ +\cdots+\frac{ \cos^{(n)}(0)}{n!} \delta x^n + o(\delta x^n)$$
$$\cos(\delta x)=\color{#ee9999}{1} \\ \color{#99ee99}{+\frac{ \cos''(0)} 2 \delta x^2} \\ \color{#9999ee}{+\frac{\cos^{(3)}(0)}{3!} \delta x^3} \\ \color{#ffff88}{+\frac{\cos^{(4)}(0)}{4!} \delta x^4} \\ +\cdots+\frac{ \cos^{(n)}(0)}{n!} \delta x^n + o(\delta x^n)$$
$$\cos(\delta x)=\color{#ee9999}{1} \\ \color{#99ee99}{-\frac{ 1} 2 \delta x^2} \\ \color{#9999ee}{+\frac{\cos^{(3)}(0)}{3!} \delta x^3} \\ \color{#ffff88}{+\frac{\cos^{(4)}(0)}{4!} \delta x^4} \\ +\cdots+\frac{ \cos^{(n)}(0)}{n!} \delta x^n + o(\delta x^n)$$
$$\cos(\delta x)=\color{#ee9999}{1} \\ \color{#99ee99}{-\frac{1} 2 \delta x^2} \\ \color{#ffff88}{+\frac{\cos^{(4)}(0)}{4!} \delta x^4} \\ +\cdots+\frac{ \cos^{(n)}(0)}{n!} \delta x^n + o(\delta x^n)$$
$$\cos(\delta x)=\color{#ee9999}{1} \\ \color{#99ee99}{-\frac{1} 2 \delta x^2} \\ \color{#ffff88}{+\frac{1}{4!} \delta x^4} \\ +\cdots+\frac{ \cos^{(n)}(0)}{n!} \delta x^n + o(\delta x^n)$$
$$\cos(\delta x)=\color{#ee9999}{1} \\ \color{#99ee99}{-\frac{1} 2 \delta x^2} \\ \color{#ffff88}{+\frac{1}{4!} \delta x^4} \\ +\cdots+\frac{(-1)^n}{(2n)!} \delta x^{2n} + o(\delta x^{2n})$$

## Formule de Taylor bis

$$f(x+\delta x)=f(x)+f'(x)\delta x\\ +\frac{f^{(2)}(x)}{2!} \delta x^2 \\ +\cdots\\ +\frac{f^{(n)}(x)}{n!} \delta x^n + \color{#ee9999}{o(\delta x^n)}$$
$$f(x+\delta x)=f(x)+f'(x)\delta x \\ +\frac{f^{(2)}(x)}{2!} \delta x^2 \\ +\cdots\\ +\frac{f^{(n)}(x)}{n!} \delta x^n + \color{#ee9999}{\frac{f^{(n+1)}(c)}{(n+1)!} \delta x^{n+1}}$$
avec $c \in (x,x+\delta x)$.

## Corollaire : Formule de Maclaurin

Si $f^{(n+1)}$ sur $[x,x+\delta x]$ est bornée,

## Développements limités classiques

$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots+\frac{x^n}{n!}+o(x^n)$$ $$\cos(x)=1-\frac{x^2}{2!}+\cdots+(-1)^n\frac{x^{2n}}{(2n)!}+o(x^{2n})$$ $$\sin(x)= x-\frac{x^3}{3!}+\cdots+(-1)^n\frac{x^{2n+1}}{(2n+1)!}+o(x^{2n+1})$$ $$\frac 1 {1-x}=1+x+x^2+\cdots+x^n+o(x^n)$$ $$(1+x)^\alpha=1+\alpha x+\frac{\alpha(\alpha-1)}{2!} x^2+\cdots+\frac{\alpha(\alpha-1)\cdots(\alpha-n)}{(n+1)!}x^{n+1}+o(x^{n+1})$$ $$\ln(1+x)=x-\frac{x^2} 2+\frac{x^3} 3 +\cdots+(-1)^{n-1} \frac{x^n} n +o(x^n)$$

## Une formule Magique

$$\frac{\pi^2} 6 = 1 + \frac 1 {2^2} + \frac 1 {3^2} + \frac 1 {4^2} + \cdots + \frac 1 {n^2} + \cdots$$ La démo d'Euler par Mathologer