L’exercice ci-dessous a été posé à l’oral de l’école polytechnique en 2022 (filière PC)
Enoncé
Soient de classe
.
Soit tel que
et
Etant donné on définit une suite par
et
A quelle condition portant sur cette suite converge-t-elle vers
?
Solution proposée
La condition cherchée est :
![Rendered by QuickLaTeX.com s](https://math-os.com/wp-content/ql-cache/quicklatex.com-d2de2f58c7b5368371c5ab418ed8bc06_l3.png)
![Rendered by QuickLaTeX.com a.](https://math-os.com/wp-content/ql-cache/quicklatex.com-86ee1849da018700d745d46e99c8091f_l3.png)
Cette condition est clairement suffisante (si alors la suite est stationnaire : tous ses termes valent
à partir d’un certain rang).
Réciproquement, supposons que la suite converge vers
et montrons que
Soit Par continuité de
en
il existe
tel que :
![Rendered by QuickLaTeX.com n_{0}\in\mathbb{N}](https://math-os.com/wp-content/ql-cache/quicklatex.com-b21ae603ccb17bbcb343044402e0ec68_l3.png)
![Rendered by QuickLaTeX.com \forall n\geqslant n_{0},\thinspace\left|x_{n}-a\right|<\eta.](https://math-os.com/wp-content/ql-cache/quicklatex.com-81e92af80b964f8a57b375fd44751d56_l3.png)
![Rendered by QuickLaTeX.com n](https://math-os.com/wp-content/ql-cache/quicklatex.com-a44d662e2fcd865f31268b1147c8a4be_l3.png)
![Rendered by QuickLaTeX.com c](https://math-os.com/wp-content/ql-cache/quicklatex.com-414b3e50638440b8b1690a5cc47116f9_l3.png)
![Rendered by QuickLaTeX.com a](https://math-os.com/wp-content/ql-cache/quicklatex.com-4b566a9f93f9d3d679507f8974d0776c_l3.png)
![Rendered by QuickLaTeX.com x_{n}](https://math-os.com/wp-content/ql-cache/quicklatex.com-2ed2eb8c22be826d2b0bdc9677124fa0_l3.png)
![Rendered by QuickLaTeX.com c\in\left]a-\eta,a+\eta\right[,](https://math-os.com/wp-content/ql-cache/quicklatex.com-8f9f44b61fe0547b7312c3732cf476dc_l3.png)
![Rendered by QuickLaTeX.com x_{n_{0}}\neq a,](https://math-os.com/wp-content/ql-cache/quicklatex.com-85d805e4c5a16886a879c48a2c17661e_l3.png)
![Rendered by QuickLaTeX.com +\infty](https://math-os.com/wp-content/ql-cache/quicklatex.com-833d378f03dc0193e9947a40b4b2ef4d_l3.png)
![Rendered by QuickLaTeX.com x_{n_{0}}=a,](https://math-os.com/wp-content/ql-cache/quicklatex.com-343baf7a679c958e3611c8de0269f92f_l3.png)
![Rendered by QuickLaTeX.com f^{n_{0}}\left(s\right)=a.](https://math-os.com/wp-content/ql-cache/quicklatex.com-1f998b52ac90a3a0cc45c94071f6517f_l3.png)
![Rendered by QuickLaTeX.com s\in\Omega.](https://math-os.com/wp-content/ql-cache/quicklatex.com-833dde8aee282af855796b83561a28af_l3.png)