Cas croissant

Aimé Lachal Suites récurrentes Cas croissant

II. Cas croissant

Exemple 1a

\(\color{blue}\;f(x)\!=\!\sqrt{x+1}\;\) \(\;u_0\!=\!1.2\;\)

Point fixe : \(\color{blue}\ell\!=\!\frac{1}{2}\!\big(1\!+\!\sqrt5\big)\approx 1.618\)

Dérivée au point fixe : \(\color{blue}\begin{array}{l}f'(\ell)\!=\!\frac{1}{4}\!\big(\sqrt5\!-\!1\big)\approx 0.309 \\ |f'(\ell)|\!<1\end{array}\)

Aperçu global

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\(u_0 < u_1\)

La suite \((u_n)_{n\in\mathbb{N}}\) est croissante
et converge vers \((1\!+\!\sqrt5)/2\)

Exemple 1b

\(\color{blue}\;f(x)\!=\!\sqrt{x\!+\!1}\;\) \(\;u_0\!=\!2\;\)

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\(u_0 > u_1\)

La suite \((u_n)_{n\in\mathbb{N}}\) est décroissante
et converge vers \((1\!+\!\sqrt5)/2\)

Exemple 2

\(\color{red}\;g(x)\!=\!x^2-1\;\) \(\;u_0\!=\!1.63\;\)

Points fixes : \(\color{red}\begin{array}{l}\ell_1\!=\!\frac{1}{2}\!\big(1\!-\!\sqrt{5}\big)\approx -0.618 \\ \ell_2\!=\!\frac{1}{2}\!\big(1\!+\!\sqrt{5}\big)\approx 1.618\end{array}\)

Dérivée aux points fixes : \(\color{red}\begin{array}{l} g'(\ell_1)\!=\!1\!-\!\sqrt{5}\approx-1.236 \\ g'(\ell_2)\!=\!1\!+\!\sqrt{5}\approx3.236 \\ |g'(\ell_1)|>1\qquad |g'(\ell_2)|>1\end{array}\)

Aperçu global

animation

stop

\(u_0 < u_1\)

La suite \((u_n)_{n\in\mathbb{N}}\) est croissante
et diverge vers \(+\infty\)

◄ Sommaire ►

Point fixe :	\(\color{blue}\ell\!=\!\frac{1}{2}\!\big(1\!+\!\sqrt5\big)\approx 1.618\)
Dérivée au point fixe :	\(\color{blue}\begin{array}{l}f'(\ell)\!=\!\frac{1}{4}\!\big(\sqrt5\!-\!1\big)\approx 0.309 \\ \|f'(\ell)\|\!<1\end{array}\)

Points fixes :	\(\color{red}\begin{array}{l}\ell_1\!=\!\frac{1}{2}\!\big(1\!-\!\sqrt{5}\big)\approx -0.618 \\ \ell_2\!=\!\frac{1}{2}\!\big(1\!+\!\sqrt{5}\big)\approx 1.618\end{array}\)
Dérivée aux points fixes :	\(\color{red}\begin{array}{l} g'(\ell_1)\!=\!1\!-\!\sqrt{5}\approx-1.236 \\ g'(\ell_2)\!=\!1\!+\!\sqrt{5}\approx3.236 \\ \|g'(\ell_1)\|>1\qquad \|g'(\ell_2)\|>1\end{array}\)