From 6d15653f80482c904b19a3b6ae25639abb903c1d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?S=C3=A9bastien=20Miquel?= Date: Sun, 3 May 2026 19:14:38 +0200 Subject: [PATCH] =?UTF-8?q?Plus=20que=20100=E2=80=A6?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Exercices 2025.org | 894 +++++++++++++++++++++++-------------- Exercices XENS MP 2025.pdf | Bin 809647 -> 664977 bytes 2 files changed, 559 insertions(+), 335 deletions(-) diff --git a/Exercices 2025.org b/Exercices 2025.org index b0ade70..e323737 100644 --- a/Exercices 2025.org +++ b/Exercices 2025.org @@ -1,8 +1,8 @@ -# -*- org-export-switch: "todoes"; -*- +# -*- org-export-switch: "xens mp"; -*- #+title: Exercices 2025 #+author: Sébastien Miquel #+date: 22-07-2025 -# Time-stamp: <03-03-26 14:07> +# Time-stamp: <03-05-26 19:14> #+OPTIONS: toc:t * Meta :noexport: @@ -14,7 +14,7 @@ #+RESULTS: | ? | ! | todo | unexed | unexed xens | -| 4 | 9 | 13 | 1477 | 139 | +| 2 | 6 | 10 | 1433 | 100 | ** Options @@ -35,10 +35,10 @@ *** XENS MP -# #+select_tags: xens -# #+exclude_tags: autre -# #+exclude_types: proof -# #+export_file_name: Exercices XENS MP 2025 +#+select_tags: xens +#+exclude_tags: autre +#+exclude_types: proof +#+export_file_name: Exercices XENS MP 2025 *** Centrale @@ -59,10 +59,10 @@ *** todoes -#+options: title:nil nopage:t tags:nil toc:nil -#+select_tags: todo -#+export_file_name: Exercices 2025 todo -#+relocate_tags: todo +# #+options: title:nil nopage:t tags:nil toc:nil +# #+select_tags: todo +# #+export_file_name: Exercices 2025 todo +# #+relocate_tags: todo *** autre @@ -1200,6 +1200,7 @@ Soient $n \ge 2$, $a: [0,1] \to \mathcal{S}_n^+(\mathbb{R})$ continue et $A = \i 6. On a la convexité pour la moitié. #+END_proof +** Analyse # ID:8734 #+begin_exercice [ENS P 2025 # 65] @@ -1250,23 +1251,29 @@ On note $E$ l'ensemble des fonctions lipschitziennes 1-périodiques de $\mathbb #+END_proof -#+begin_exercice [ENS P 2025 # 68] :todo: +# ID:8838 +#+begin_exercice [ENS P 2025 # 68] Soient $E$ l'espace des suites réelles $(x_n)_{n\geq 0}$ nulles à partir d'un certain rang, et $T\in\mathcal{L}(E)$. On suppose $T$ continu pour la norme $\|\ \|_1$ et pour la norme $\|\ \|_{\infty}$. Montrer que $T$ est continue pour la norme $\|\ \|_2$. #+end_exercice -#+BEGIN_proof :todo: -On considère une suite de suites $(u^n)$ dont la norme $2$ tend vers $0$. En particulier, la norme infinie tend vers $0$, donc $T(u^n)$ tend, en norme infinie, vers $0$. +#+BEGIN_proof +Notons $C_1$ et $C_\infty$ les normes de $T$ pour les normes respectives $\|\cdot\|_1$ et $\|\cdot\|_\infty$. -La fonction $T$ est lipschitzienne pour les deux normes. +Soit $(e_j)_{j\in\N}$ la base de $E$ et $a_{i,j} = (T(e_j))_i$. -Si on suppose $\lN u^n\rN_1 \ra 0$, alors $\lN T(u^n)\rN_1\ra 0$, ce qui implique $\lN T(u^n)\rN_2 \ra 0$. +Pour tout $j \in \N$, on a $\|T(e_j)\|_1 = \sum_{i=0}^\infty |a_{i,j}| \leq C_1$. -Si on suppose $\lN u^n\rN_1$ bornée. On a $\lN T(u^n)\rN_2^2 \leq \lN -T(u^n)\rN_{\i} \lN T(u^n)\rN_1\ra 0$. +Fixons $i \in \N$. Pour tout ensemble fini $J \subset \N$, posons $x = \sum_{j \in J} \mathrm{sgn}(a_{i,j}) e_j \in E$. On a $\|x\|_\infty \leq 1$, d'où : +$$T(x)_i = \sum_{j \in J} |a_{i,j}| \leq \|T(x)\|_\infty \leq C_\infty.$$ +Comme cela est vrai pour tout sous-ensemble fini $J$, on en déduit que $\sum_{j=0}^\infty |a_{i,j}| \leq C_\infty$. -En général : on découpe la suite pour regrouper en des groupes de -termes dont la norme $1$ vaut $1$. (Il y a un nombre fini de groupes à -chaque fois car les suites sont nulles APCR). +Soit maintenant $x \in E$. Évaluons $\|T(x)\|_2^2$ en utilisant l'inégalité de Cauchy-Schwarz : +$$(T(x)_i)^2 = \left( \sum_{j=0}^\infty a_{i,j} x_j \right)^2 \leq \left( \sum_{j=0}^\infty |a_{i,j}| \right) \left( \sum_{j=0}^\infty |a_{i,j}| x_j^2 \right) \leq C_\infty \sum_{j=0}^\infty |a_{i,j}| x_j^2.$$ +(Les sommes sur $j$ sont en fait finies car $x \in E$ est nulle à partir d'un certain rang). +En sommant sur $i$, on obtient par interversion des sommes (les termes étant positifs) : +$$\|T(x)\|_2^2 = \sum_{i=0}^\infty (T(x)_i)^2 \leq C_\infty \sum_{i=0}^\infty \sum_{j=0}^\infty |a_{i,j}| x_j^2 = C_\infty \sum_{j=0}^\infty x_j^2 \left( \sum_{i=0}^\infty |a_{i,j}| \right) \leq C_1 C_\infty \sum_{j=0}^\infty x_j^2 = C_1 C_\infty \|x\|_2^2.$$ + +Ainsi, $T$ est continu pour la norme $\|\cdot\|_2$, avec une norme subordonnée majorée par $\sqrt{C_1 C_\infty}$. #+END_proof @@ -1399,17 +1406,40 @@ On trouve si et seulement si $\sum \frac{1}{n_k}$ diverge. Pour $\lN \cdot\rN_{\i}$, il faut $n_0 = 0$. Il faut aussi que $\sum \frac{1}{n_k}$ diverge. Réciproquement, en utilisant $\lN f-g\rN_{\i}\leq \lN f'-g'\rN_2$, on peut conclure. #+END_proof - - -#+BEGIN_exercice [ENS L 2025 # 77] :todo: +# ID:8839 +#+BEGIN_exercice [ENS L 2025 # 77] Soit $f \in \mathcal{C}^0(\mathbb{R}, \mathbb{R})$. On note $D = \{\ell 2^{-k} + 2^{-k}[0, 1] ; (k, \ell) \in \mathbb{Z}^2\}$. Pour tout intervalle $I$ de $D$, on note $\op{long}(I)$ la longueur de $I$ et on pose $M_I(f) = \frac{1}{\op{long}(I)} \int_I f$. On - pose $||f|| = \sup \left\{ \frac{1}{\op{long}(I)} \int_{\Gamma} |f M_I(f)| \; ; \; I \in D \right\}$. - 1. On suppose $||f||$ finie. Soit $m \in \N^*$, $(I, J) \in D^2$ avec $I \subset J$ tels que $\op{long}(J) = 2^m \op{long}(I)$. Démontrer que $|M_I(f)- M_J(f)| \le 2m||f||$. - 1. On suppose que ||f|| = 1 et $M_{[0,1]}(f) = 0$. - On note $F_k = \{I \in D : I \subset [0,1], M_I(f) > 5k \text{ et } I \text{ maximal pour cette propriété}\}$. On pose $\Omega_k = \bigcup_{I \in F_k} I \text{ et et } \op{long}(\Omega_k) = \sum_{I \in F_k} \op{long}(F_I)$. + pose $\|f\| = \sup \left\{ \frac{1}{\op{long}(I)} \int_{\Gamma} |f - M_I(f)| \; ; \; I \in D \right\}$. + 1. On suppose $\|f\|$ finie. Soit $m \in \N^*$, $(I, J) \in D^2$ avec $I \subset J$ tels que $\op{long}(J) = 2^m \op{long}(I)$. Démontrer que $|M_I(f)- M_J(f)| \le 2m\|f\|$. + 1. On suppose que $\|f\|$ = 1 et $M_{[0,1]}(f) = 0$. + On note $F_k = \{I \in D : I \subset [0,1], M_I(f) > 5k \text{ et } I \text{ maximal pour cette propriété}\}$. On pose $\Omega_k = \bigcup_{I \in F_k} I \text{ et } \op{long}(\Omega_k) = \sum_{I \in F_k} \op{long}(F_I)$. Montrer que, pour $k \geq 1$, $\op{long}(\Omega_k) \leqslant \frac{1}{3} \op{long}(\Omega_{k-1})$. #+END_exercice +#+BEGIN_proof + 1. Il suffit de traiter le cas $m = 1$. On peut supposer que $M_J(f) = 0$. + + Alors $|M_I(f)| \leq \frac{1}{|I|}\int_I |f|$ + et $\lN f\rN \leq \frac{1}{|J|} \int_J |f|$, donc $|M_I(f)|\leq 2 \lN f\rN$. + 2. Les intervalles étant dyadiques, ilssont soit disjoints, soit inclus l'un dans l'autre. + + De plus, comme $M_{[0,1]}(f) = 0 \leq 5k$, $[0,1]$ n'est pas dans $F_k$. Tout $J \in F_k$ admet donc un intervalle parent $\tilde{J} \in D$ tel que $J \subset \tilde{J} \subset [0,1]$ et $\op{long}(\tilde{J}) = 2\op{long}(J)$. + Par maximalité de $J$, on a $M_{\tilde{J}}(f) \leq 5k$. D'après la question précédente, $M_J(f) \leq M_{\tilde{J}}(f) + 2\|f\| \leq 5k + 2$. + + Pour $k \geq 1$, si $I \in F_k$, on a $M_I(f) > 5k > 5(k-1)$. Il existe donc un (unique) intervalle maximal $J \in F_{k-1}$ tel que $I \subset J$. On peut ainsi partitionner $F_k$ selon l'intervalle $J \in F_{k-1}$ qui contient ses éléments. + + Fixons $J \in F_{k-1}$ et posons $S_J = \bigcup_{I \in F_k, I \subset J} I$. On évalue l'intégrale sur $S_J$ : + $$\int_{S_J} (f - M_J(f))\dx = \sum_{I \subset J} \int_I (f - M_J(f))\dx = \sum_{I \subset J} \op{long}(I)(M_I(f) - M_J(f)).$$ + + Or, pour $I \in F_k$, $M_I(f) > 5k$, et d'après ce qui précède, $M_J(f) \leq 5(k-1) + 2 = 5k - 3$. Ainsi, $M_I(f) - M_J(f) > 3$. + On en déduit que $\int_{S_J} (f - M_J(f))\dx > 3 \sum_{I \subset J} \op{long}(I) = 3 \op{long}(S_J)$. + + D'autre part, en utilisant $\|f\| = 1$ : + $$\int_{S_J} (f - M_J(f))\dx \leq \int_{S_J} |f - M_J(f)|\dx \leq \int_J |f - M_J(f)|\dx \leq \op{long}(J)\|f\| = \op{long}(J).$$ + Par conséquent, $3\op{long}(S_J) < \op{long}(J)$. + En sommant cette inégalité sur tous les intervalles $J \in F_{k-1}$, on obtient bien : + $$\op{long}(\Omega_k) = \sum_{J \in F_{k-1}} \op{long}(S_J) \leq \frac{1}{3} \sum_{J \in F_{k-1}} \op{long}(J) = \frac{1}{3}\op{long}(\Omega_{k-1}).$$ +#+end_proof # ID:8780 @@ -1422,7 +1452,8 @@ On munit les espaces $\ell^1_{\mathbb{Z}}(\mathbb{R})$ et $\ell^2_{\mathbb{Z}}(\ 1. Soit $u \in \mathbb{R}^{\mathbb{Z}}$. Montrer que $u \in H$ si et seulement si l'application $\mu_u : H \to H$ définie par$\forall v \in H, \ \mu_u(v) = u * v \text{ avec } (u * v)_n = \sum_{i=1}^n u_i v_i \text{ est bien définie et continue.}$ #+end_exercice -#+begin_exercice [ENS P 2025 # 79] :todo: +# ID:8840 +#+begin_exercice [ENS P 2025 # 79] On note $\ell^1$ l'ensemble des suites sommables de $\mathbb{R}^{\N}$. On munit $\ell^1$ de la norme définie, pour $u = (u_n)_{n \geq 0}$, par $||u||_1 = \sum_{n=0}^{+\infty} |u_n|$. Soient $(u^k)_{k\in\N}$ une suite d'éléments de $\ell^1$ et $u\in\ell^1$. Montrer l'équivalence entre : + la suite $(u^{\overline{k}})_{k\in\N}$ converge vers $u$ pour la norme $\| \|_1$, @@ -1430,7 +1461,12 @@ Soient $(u^k)_{k\in\N}$ une suite d'éléments de $\ell^1$ et $u\in\ell^1$. Mont #+end_exercice #+BEGIN_proof $i)\Rightarrow ii)$ est trivial - Réciproquement, on veut prendre $\phi_n = \sign (u_n^k - u_n)$, s'il est fixé à partir d'un certain rang. Si ce n'est pas le cas, alors forcément $u_n = 0$ et la somme de ces termes tend vers $0$ par CVD ?? !! Non. + +Réciproquement, on a $u^k_n \tend{k\ra +\i} u_n$. + +Extraction diagonale : on peut supposer que $\sign (u_0^k - u_0)$ est +constant, puis que $\sign u_1^k -u_1$ est constant etc. Puis on prend +cette suite de signe. On obtient que toute suite extraite admet une suite extraite qui converge pour $\lN \rN_1$, d'où le résultat. #+END_proof @@ -1448,24 +1484,52 @@ On note $S=\{z\in\C,\,|z|=1\}$ et $\Gamma=\{\gamma\in\mathcal{C}^0([0,1],S)\,\,; #+END_proof -#+BEGIN_exercice [ENS PLSR 2025 # 81] :todo: +# ID:8841 +#+BEGIN_exercice [ENS PLSR 2025 # 81] 1. Soit $f \in \mathcal{C}^2(\mathbb{R}, \mathbb{R})$ telle qu'il existe $x^* \in \mathbb{R}$ vérifiant $f(x^*) = 0$ et $f'(x^*) \neq 0$. - On définit par récurrence une suite $(x_k)$ avec $x_0 \in \mathbb{R}$ et $x_{k+1} = x_k \frac{f(x_k)}{f'(x_k)}$. + On définit par récurrence une suite $(x_k)$ avec $x_0 \in \mathbb{R}$ et $x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}$. - Montrer qu'il existe $\eps > 0$ tel que, pour $x_0 \in [x^* \eps, x^* + \eps]$, la suite $(x_k)$ est bien définie et converge vers $x^*$. + Montrer qu'il existe $\eps > 0$ tel que, pour $x_0 \in [x^* -\eps, x^* + \eps]$, la suite $(x_k)$ est bien définie et converge vers $x^*$. 2. Avec $f\colon x \mapsto e^x- 1$, quelles sont les valeurs de $x_0 \in \mathbb{R}$ pour lesquelles la suite $(x_k)$ précédente est stationnaire? 3. On revient au cas général et on suppose $f'' > 0$ et $f'$ ne s'annule pas sur $\mathbb{R}$. Pour quelles valeurs de $x_0 \in \mathbb{R}$ la suite $(x_k)$ est-elle stationnaire? #+END_exercice +#+BEGIN_proof + 1. La fonction $g\colon x\mapsto x-\frac{f(x)}{f'(x)}$ vérifie $g'(x^*) = 0$. + 2. + 3. Seulement pour le point fixe. +#+END_proof -#+begin_exercice [ENS L 2025 # 82] :todo: -Soit $f \in \mathcal{C}^0([a,b],[a,b])$. On suppose dans les questions 1) et 2) que $f$ n'a pas de - point de période 2, c'est-à-dire que $\forall x \in [a,b], f(x) \neq x \Rightarrow (f \circ f)(x) \neq x$. +# ID:8842 +#+begin_exercice [ENS L 2025 # 82] +Soit $f \in \mathcal{C}^0([a,b],[a,b])$. On suppose dans les deux +premières questions que $f$ n'a pas de point de période 2, +c'est-à-dire que $\forall x \in [a,b], f(x) \neq x \Rightarrow (f +\circ f)(x) \neq x$. 1. Soit $c \in [a, b]$ tel que $f(c) >c$. Montrer que pour tout $k \in \N^*$, $f^k(c) > c$. - 1. Soit $x_0 \in [a,b]$, on pose pour tout $n$, $x_{n+1} = f(x_n)$. Démontrer que la suite $(x_n)$ - converge. - 1. Démontrer que la suite $(x_n)$ converge pour tout choix de $x_0$ si et seulement si $f$ n'a pas de point de période 2. + 2. Soit $x_0 \in [a,b]$, on pose pour tout $n$, $x_{n+1} = f(x_n)$. + Démontrer que la suite $(x_n)$ converge. + 3. Démontrer que la suite $(x_n)$ converge pour tout choix de $x_0$ si et seulement si $f$ n'a pas de point de période 2. #+end_exercice +#+begin_proof + 1. Soit $c$ tel que $f(c) > c$. Supposons que $f^2(c)\leq c$, + c'est-à-dire $f^2(c)\lt c$. On va chercher un point fixe de $f^2$ + à gauche de $c$. + + Notons que $f^2(a)\geq a$. Il y a donc un changement de signe de $f^2(x) - x$ à gauche de $c$. On peut considérer le premier $0$. + + Si c'était un point fixe, on aurait $f(x)\geq x$ à droite de celui-ci, donc $f^2(x)\geq x$ au voisinage à droite de ce point fixe, contradiction. + + Idem pour $k$ quelconque. + + De même, si $f(c)\lt c$. Si on avait $f^2(c)\gt c$, en regardant à droite, on aurait une contradiction. + 2. Si $x_{n+1}\geq x_n$, alors $\forall k,\, x_{n+k}\geq x_n$. Les + instants croissants, donnent des bornes inférieures. Idem pour les + instants décroissants une borne supérieure, et les deux doivent + vérifier $f(\a) =\b$ et $f(\b) = \a$. + 3. Trivial. +#+end_proof + # ID:nil # Classique #+begin_exercice [ENS PLSR 2025 # 83] @@ -1487,21 +1551,22 @@ Simple. #+END_proof -#+begin_exercice [ENS L 2025 # 85] :todo: +# ID:8843 +#+begin_exercice [ENS L 2025 # 85] 1. Pour quels réels $s$ la somme $\sum_{n\neq m\in\N^*} \frac{|n-m|^s}{nm(n^2-m^2)^2}$ est-elle finie? 2. E Pour $n=(n_1,n_2)\in \mathbb{Z}^2$ on, on note $|n|=\sqrt{n_1^2+n_2^2}$. 3. Pour quels réels $s$ la somme $\sum_{(n,m)\in (\mathbb{Z}^2\backslash \{0\})^2} \frac{|n-m|^s}{|n||m|(1+(|n|-|m|)^2)}$ est-elle finie ? #+end_exercice -#+BEGIN_proof :todo: +#+BEGIN_proof 1. On peut prendre $n\gt m$. Poser $u = n-m$ et sommer sur $u$ et $m$ : $\sum_{u,m\geq 1}\frac{u^s}{m (m+u) u^2 (2 m+ u)}$. Pour $u$ fixé, on trouve facilement un équivalent de la somme. 2. - 3. Sommer sur $m$, puis sur les $n$ tel que $|n-m|\in [k, k+4]$. Ce - sont les $n$ dans un anneau $C_k$. On somme $\sum_{n\in C_{k,m}} \frac{1}{|n|} \frac{1}{1 + (|n|- |m|)^2}$. On peut par ailleurs supposer que $|n|\geq |m|$. - - Il semble plus simple de prendre la norme infinie en haut, les $n\in C_{k,m}$ sont clairs, ils sont sur les quatre bords d'un carré. Pour le bord de droite, … + 3. Cela diverge car si on ne considère que les termes pour lesquels $n\simeq m$ (à une distance d'au plus $2$), si on somme sur $K = |n|$ (à $\pm 1$ près) on obtient + $\sum_{K} \sum_{n\simeq m, |n| = K} \frac{1}{K^2} = \sum_{K} \frac{1}{K}$. #+END_proof + + # ID:8788 #+begin_exercice [ENS PLSR 2025 # 86] On note $S$ l'ensemble des suites croissantes à termes dans $\N \setminus \{0,1\}$. @@ -1518,10 +1583,28 @@ On note $S$ l'ensemble des suites croissantes à termes dans $\N \setminus \{0,1 #+END_proof -#+begin_exercice [ENS L 2025 # 87] :todo: -Soit $f: \N \to \mathbb{R}^{+*}$ décroissante de limite nulle. Soit $\phi: \N \to \N$ croissante. On suppose que, pour tout $\alpha \in \mathbb{R}^{+*}$, il existe une unique suite $(n_i)_{i \in \N}$ telle que $\alpha = \sum_{i=0}^{+\infty} f(n_i)$ et, pour tout $i \in \N$, $n_{i+1} \geq \phi(n_i)$. - Montrer que $\phi(0)=0$ et, pour tout $n\in\N^*$, $f(n-1)=\sum_{i=0}^{+\infty}f\left(\phi^i(n)\right)$, où $\phi^i$ désigne l'itérée $i$-ème de $\phi$ pour la composition des applications. +# ID:8848 +#+begin_exercice [ENS L 2025 # 87] +Soit $f\colon \N \to \mathbb{R}^{+*}$ décroissante de limite nulle. Soit $\phi\colon \N \to \N$ croissante. On suppose que, pour tout $\alpha \in \mathbb{R}^{+*}$, il existe une unique suite $(n_i)_{i \in \N}$ telle que $\alpha = \sum_{i=0}^{+\infty} f(n_i)$ et, pour tout $i \in \N$, $n_{i+1} \geq \phi(n_i)$. + +Montrer que $\phi(0)=0$ et, pour tout $n\in\N^*$, $f(n-1)=\sum_{i=0}^{+\infty}f\left(\phi^i(n)\right)$, où $\phi^i$ désigne l'itérée $i$-ème de $\phi$ pour la composition des applications. #+end_exercice +#+BEGIN_proof +Si $\phi(0) = 1$ par exemple + +Certains $\a$ s'écrivent avec $f(0)$, d'autre non. Si une suite s'écrit sans $f(0)$, alors $\a + f(0)$ s'écrit avec $f(0)$. Si une suite s'écrit avec $f(0)$, alors $\a + f(0)$ doit s'écrire sans $f(0)$. + +Alors tous les $\a\in ]0, f(0)]$ s'écrivent sans $f(0)$, puis les autres avec jusqu'à $2f(0)$, etc. + +Mais si on prend $2f(0) = \sum f(n_i)$ (avec $f(0)$). Si à un instant $n_i \gt \phi(n_i)$, on peut décroître $n_i$. Ou bien cela garde la même valeur de $f$, ce qui contredit l'unicité, ou bien la valeur de $f$ augment strictement, mais de moins de $f(0)$, donc cela contredit le fait que les $\a\in \interval]{2f(0), 3f(0)}]$ s'écrivent sans $f(0)$. + +Si $\phi(0) = 2$, on peut avoir $f(0)$, ou $f(1)$, ou aucun. Si aucun, $\a + f(0)$ s'écrit avec $f(0)$. + +Il y a tout un intervalle de «aucun». Puis $f(0)+ \text{aucun}$ donne du $f(0)$. + +Mais $f(0)$ doit s'écrire sans $f(0)$, alors que $f(0)+\eps$ s'écrit avec, ce qui donne une contradiction comme précédemment. +#+END_proof + # ID:8744 #+begin_exercice [ENS P 2025 # 88] @@ -1562,12 +1645,23 @@ Soit $f\colon \mathbb{R} \to \mathbb{R}$ telle que $\sum f(a_n)$ converge pour t Trouver les $f\colon [0,1] \ra \R$ continues telles que $\forall x \in [0,1], \ f(x) = \sum_{n=0}^{+\i} \frac{f(x^n)}{2^n}$. #+end_exercice -#+begin_exercice [ENS PLSR 2025 # 92] :todo: +# ID:8861 +#+begin_exercice [ENS PLSR 2025 # 92] Soit $f$ une fonction de $\R$ dans $\R \cup \{+\i\}$ non identiquement égale à $+\i$. Pour $y \in \R$, on pose $f^*(y) = \sup\{xy - f(x) ; x \in \R\}$. 1. Montrer que $\{x \in \R, f^*(x) \lt +\i\}$ est un intervalle (éventuellement vide) sur lequel $f^*$ est convexe. - 1. Montrer que, si $f$ est dérivable et convexe sur $\R$, alors $f^{*} = f$. + 1. Montrer que, si $f$ est dérivable et convexe sur $\R$, alors $\big(f^{*}\big)^* = f$. 1. On suppose que $f$ est de classe $C^2$ sur $\R$, que $f''\gt 0$ sur $\R$ et que $\frac{f(x)}{|x|}\underset{|x|\ra+\i}{\longrightarrow}+\i$. Montrer que $f^*$ est dérivable sur $\R$ et que : $\forall (x,y)\in\R^2,\,y=f'(x)\Leftrightarrow x=(f^*)'(y)$. #+end_exercice +#+BEGIN_proof + 1. Si $x\mapsto xy_1 - f(x)$ est majorée et $x\mapsto xy_2 - f(x)$ est majorée, clairement pour tout $y\in [y_1, y_2]$ $x\mapsto xy -f(x)$ est majorée, et $f^*$ est convexe. + 2. On a $f^*(y)\geq xy - f(x)$, donc $f(x)\geq xy - f^*(y)$, en passant au sup sur $y$, on obtient $f(x)\geq f^{**}(x)$. + + Soit $x_0$. On pose $y_0 = f'(x_0)$, alors $f(x)\geq f'(x_0) (x-x_0) + f(x_0)$ + + On en déduit que $f^*(y_0) = x_0 y_0 - f(x_0)$, puis que $f^{**}(x_0)\geq f(x_0)$. + 3. +#+END_proof + # ID:8781 #+begin_exercice [ENS SR 2025 # 93] @@ -1592,11 +1686,29 @@ Peut-on l'étendre aux puissances fractionnaires ? Oui : $f(x)^{1/q} = f(x+\ln_2 #+END_proof -# Relier à 6558 -#+begin_exercice [ENS PLSR 2025 # 95] :todo: +# ID:8844 +#+begin_exercice [ENS PLSR 2025 # 95] 1. Soient $a, b \in \R$ avec $|b| \lt \pi$. Montrer qu'il existe $z \in \C$ tel que $z + e^z = a + ib$. 1. Montrer que l'application $z \mapsto z e^z$ est surjective de $\C$ sur $\C$. #+end_exercice +#+begin_proof + 1. En posant $z = x+iy$, l'équation $z + e^z = a+ib$ se réécrit en identifiant ses parties réelle et imaginaire : + $$\left\{\begin{aligned}x + e^x \cos y &= a\\ y + e^x \sin y &= b\end{aligned}\right.$$ + Si $b = 0$ on est bon. + + Si $b \in \interval]{0, \pi}[$, on cherche une solution avec $y \in \interval]{0, \pi}[$ : on veut $e^x = \frac{b-y}{\sin y}$. En injectant cette relation dans la première équation, on obtient : + $$g(y) = \ln\left(\frac{b-y}{\sin y}\right) + (b-y)\cot y = a$$ + En considérant les limite de $g$ en $0^+$ et $b^{-}$, on a une solution. + 2. Soit $w \in \C$. On cherche $z \in \C$ tel que $z e^z = w$. + + Si $w \in \C \setminus (-\infty, 0]$, on peut l'écrire sous forme exponentielle $w = e^{a+ib}$ avec $a \in \R$ et $b \in (-\pi, \pi)$. D'après la question 1, il existe $Z \in \C$ tel que $Z + e^Z = a+ib$. En posant $z = e^Z$, on obtient $z e^z = e^Z e^{e^Z} = e^{Z+e^Z} = e^{a+ib} = w$. + + Si $w = 0$, $z = 0$ est solution. + + Si $w \in [-1/e, 0)$, on peut prendre $z = \in \R$. + + Si $w \in (-\infty, -1/e)$, on cherche $z$ sous la forme $x+iy$ avec $y \in \interval]{0, \pi}[$. L'équation s'écrit $e^x(x \cos y - y \sin y) + i e^x(x \sin y + y \cos y) = w$. + En annulant la partie imaginaire, on obtient $x = -y \cot y$. En remplaçant $x$ dans la partie réelle, on doit résoudre : + $$e^{-y \cot y} \left(-y \frac{\cos^2 y}{\sin y} - y \sin y\right) = -\frac{y}{\sin y} e^{-y \cot y} = w$$ + par limites. +#+end_proof + # ID:8746 #+begin_exercice [ENS P 2025 # 96] @@ -1688,7 +1800,8 @@ Puis on peut encadrer $f$ par de telles fonctions. #+END_proof -#+begin_exercice [ENS SR 2025 # 103] :todo: +# ID:nil # Chiant. +#+begin_exercice [ENS SR 2025 # 103] Soient $N \in \N^*$ et $(x_1, \dots, x_N) \in \C^N$. Pour $y \in \R$, on note $e(y) = e^{2i\pi y}$. Soit $f\colon t \in \R \mapsto \sum_{n=1}^N x_n e(nt)$. Soient $R \in \N^*$ et $(t_1, \dots, t_R) \in \R^R$. @@ -1696,7 +1809,9 @@ Soit $f\colon t \in \R \mapsto \sum_{n=1}^N x_n e(nt)$. Soient $R \in \N^*$ et $ 1. 1. Montrer que $\sum_{r=1}^R |f(t_r)|^2 \leq NR \sum_{k=1}^N |x_k|^2$. 2. Étudier le cas d'égalité dans l'inégalité précédente. - 2. Pour $t \in \R$, on pose $\Delta(t) = \inf_{n \in \Z} |n - t|$. On suppose les $t_i$ distincts. Soit $\delta \gt 0$ tel que $\delta \leq \min\limits_{1\leq i\neq j\leq R} \Delta(t_i-t_j)$. Montrer que $\sum_{r=1}^R |f(t_r)|^2 \leq (2N\pi+\delta^{-1}) \sum_{r=1}^N |x_k|^2$. + 2. Pour $t \in \R$, on pose $\Delta(t) = \inf_{n \in \Z} |n - t|$. On suppose les $t_i$ distincts. + + Soit $\delta \gt 0$ tel que $\delta \leq \min\limits_{1\leq i\neq j\leq R} \Delta(t_i-t_j)$. Montrer que $\sum_{r=1}^R |f(t_r)|^2 \leq (2N\pi+\delta^{-1}) \sum_{r=1}^N |x_k|^2$. Ind. On pourra montrer que, pour une fonction $g$ de classe $\mc C^1$ sur $\R$, pour $a \in \R$ et $h \gt 0$, $$|g(a)| \le \frac{1}{2h} \int_{a-h}^{a+h} |g(t)| dt + \frac{1}{2} \int_{a-h}^{a+h} |g'(t)| dt$$. @@ -1722,11 +1837,19 @@ On note $E$ l'ensemble des fonctions 1-périodiques et de classe $\mc C^{\i}$ de #+END_proof -#+begin_exercice [ENS P 2025 # 105] :todo: -Soient $a,b\gt 0$ et $m\in\Z$. Calculer $I_m(a,b)=\int_a^{+\i}e^{-ax-\frac{b}{x}}x^{m-\frac{1}{2}}\dx$. +# ID:8872 +#+begin_exercice [ENS P 2025 # 105] +Soient $a,b\gt 0$ et $m\in\Z$. Calculer $I_m(a,b)=\int_0^{+\i}e^{-ax-\frac{b}{x}}x^{m-\frac{1}{2}}\dx$. #+end_exercice #+BEGIN_proof +Pour $I_0$ poser $x = t^2$, on obtient $2\int e^{-at^2 - \frac{b}{t^2}}\dt$. On peut +regrouper $a$ et $b$, puis dériver par rapport au paramètre en +question. On obtient une équation différentielle. +Pour $m\in\N$, c'est obtenu en dérivant l'expression d'origine +par rapport à $a$. + +Pour $m$ négatif, poser $u = \frac{1}{x}$ peut-être. #+END_proof @@ -1740,29 +1863,48 @@ Il faut $\lN e^{t^2 A}\rN\ra 0$, donc les valeurs propres de $A$ ont des parties Réciproquement, dans ce cas c'est bon. car $e^A$ est tri sup avec des coefficients diagonaux $\lt 1$, donc ses puissances tendent vers $0$ de manière exponentielle. #+END_proof - -#+begin_exercice [ENS PLSR 2025 # 107] :todo: +# ID:8870 +#+begin_exercice [ENS PLSR 2025 # 107] Soit $f\colon \R \ra \R$ lipschitzienne. On suppose qu'il existe $R \gt 0$ tel que, pour tout $x \in \R \setminus [-R, R]$, $f(x) = 0$. - 1. Montrer que $\eps \mapsto \int_{-\i}^{-\eps} \frac{f(x)}{x} dx + \int_{\eps}^{+\i} \frac{f(x)}{x} dx$ admet une limite en $0^+$. -On note $\op{vp}\left(\int_{-\i}^{+\i} \frac{f(x)}{x} dx\right)$ cette limite. - 1. On note $T_f\colon x \mapsto \int_{-\i}^x f(y) \ln |y- x| dy + \int_x^{+\i} f(y) \ln |y- x| dy$. Justifier que $T_f$ est bien définie sur $\R$. + 1. Montrer que $\eps \mapsto \int_{-\i}^{-\eps} \frac{f(x)}{x} \dx + \int_{\eps}^{+\i} \frac{f(x)}{x} \dx$ admet une limite en $0^+$. + 2. E On note $\op{vp}\left(\int_{-\i}^{+\i} \frac{f(x)}{x} \dx\right)$ cette limite. + 1. On note $T_f\colon x \mapsto \int_{-\i}^x f(y) \ln |y- x| \dy + \int_x^{+\i} f(y) \ln |y- x| \dy$. Justifier que $T_f$ est bien définie sur $\R$. 1. On suppose $f$ de classe $C^1$. Montrer que $T_f$ est dérivable sur $\mathbb R$ et que : - $$\forall x \in \R, \ (T_f)'(x) = \op{vp}\left(\int_{-y}^{+\i} \frac{f(y+x)}{y} dy\right)$$. + $$\forall x \in \R, \ (T_f)'(x) = \op{vp}\left(\int_{-\i}^{+\i} \frac{f(y+x)}{y} dy\right)$$. #+end_exercice -#+BEGIN_proof :todo: +#+BEGIN_proof 1. Si $f(0) = 0$ c'est bon, et si $f(0)\neq 0$, on peut le lui retirer, c'est bon aussi. - 2. + 2. vide + 3. $T_f(x) = \int_{-\i}^{+\i} f(x+u) \ln|u| \du$. On peut dériver + sous le signe intégral. + + Pour faire apparaître la valeur principale, on intègre par partie sur $\R\setminus [-\eps,\eps]$. #+END_proof - -#+begin_exercice [ENS SR 2025 # 108] :todo: - 1. Pour $(p,k) \in \N^2$, montrer la convergence de $I_{p,k} = \int_0^1 \int_0^1 \frac{y^k x^p}{1 xy} \, dx \, dy$ et l'exprimer sous forme de la somme d'une série numérique. - 1. On note $d_n = \op{ppcm}(1, \dots, n)$ pour $n \in \N^*$. Montrer que $I_{p,k} \in \frac{1}{d_p^2} \Z$ si $p$ \gt $k$, et $I_{p,p} \in \zeta(2) + \frac{1}{d^2} \Z$. - 1. On pose $P_n = \frac{1}{n!} D^n (X^n (1 X)^n)$. Montrer que $P_n$ est à coefficients entiers. - 1. Montrer que $I_n = \int_0^1 \int_0^1 \frac{(1-y)^n P_n(x)}{1-xy} dx dy$ converge, et en donner une expression simplifiée. - 1. Montrer que $I_n \in \frac{1}{d^2} (\Z + \zeta(2)\Z)$. +# ID:8868 +#+begin_exercice [ENS SR 2025 # 108] + 1. Pour $(p,k) \in \N^2$, montrer la convergence de $I_{p,k} = \int_0^1 \int_0^1 \frac{y^k x^p}{1- xy} \dx \dy$ et l'exprimer sous forme de la somme d'une série numérique. + 2. On note $d_n = \op{ppcm}(1, \dots, n)$ pour $n \in \N^*$. Montrer que $I_{p,k} \in \frac{1}{d_p^2} \Z$ si $p$ \gt $k$, et $I_{p,p} \in \zeta(2) + \frac{1}{d^2} \Z$. + 3. On pose $P_n = \frac{1}{n!} D^n (X^n (1- X)^n)$. Montrer que $P_n$ est à coefficients entiers. + 4. Montrer que $I_n = \int_0^1 \int_0^1 \frac{(1-y)^n P_n(x)}{1-xy} \dx \dy$ converge, et en donner une expression simplifiée. + 5. Montrer que $I_n \in \frac{1}{d_n^2} (\Z + \zeta(2)\Z)$. + 6. S On admet que $d_n \sim e^n$. En déduire que $\zeta(2)\not\in\Q$. #+end_exercice +#+begin_proof +1. $I_{p,k} = \sum_{n=0}^\infty \int_0^1 x^{n+p} \dx \int_0^1 y^{n+k} \dy = \sum_{n=0}^\infty \frac{1}{(n+p+1)(n+k+1)}$ +2. Si $p > k$ : éléments simples + télescopage. +3. +4. En intégrant par parties $n$ fois par rapport à $x$, + $$\int_0^1 \frac{P_n(x)}{1-xy} \dx = \frac{1}{n!} \int_0^1 x^n(1-x)^n D_x^n \left( \frac{1}{1-xy} \right) \dx = \int_0^1 \frac{x^n(1-x)^n y^n}{(1-xy)^{n+1}} \dx$$ + donc + $$I_n = \int_0^1 \int_0^1 \frac{x^n(1-x)^n y^n(1-y)^n}{(1-xy)^{n+1}} \dx \dy$$ +5. Par linéarité à partir de l'expression initiale de $I_n$. +6. L'expression de $I_n$ doit tendre vers $0$ de manière géométrique, avec $\a = \sup \frac{x (1-x) y (1-y)}{1-xy}$. + + Le maximum doit être atteint sur la diagonale. +#+end_proof + # ID:8773 #+begin_exercice [ENS L 2025 # 109] @@ -1796,7 +1938,7 @@ limites. Sinon, on converge vers $F(ax + b)$, avec $a\neq 0$. #+END_proof -#+begin_exercice [ENS L 2025 # 111] +#+begin_exercice [ENS L 2025 # 111] :todo: Soit $(f_n)_{n\in\N}$ une suite de fonctions de $[0,1]$ dans $]0,1]$, convergeant simplement vers une fonction $f$. 1. Pour $n \ge 2$, on pose $t_n = \frac{1}{\ln n} \sum_{i=1}^n @@ -1817,13 +1959,15 @@ convergeant simplement vers une fonction $f$. #+end_exercice #+BEGIN_proof 1. Sans difficultés. - 2. + 2. La condition sur $\sigma_n(1/2)$ assure $f(1/2) = 0$. #+END_proof +# ID:8871 #+begin_exercice [ENS P 2025 # 112] Soit $f\colon x \mapsto \sum_{n=1}^{+\i} \frac{1}{n} \sin\left(\frac{x}{4^n}\right)$. - 1. Montrer que $\lim\limits_{x \ra +\i} (\inf \{ f(t), t \ge x \}) = 0$. + 1. Montrer que $\lim\limits_{x \ra +\i} (\inf \{ |f(t)|, t \ge x \}) = 0$. + 1. Déterminer $\lim\limits_{x \ra +\i} (\inf \{ f(t), t \ge x \})$. 1. Montrer que $0 \lt \lim\limits_{x \ra +\i} \left( \sup \left\{ \frac{|f(t)|}{\ln(\ln t)}, \ t \geq x \right\} \right) \lt +\i$. #+end_exercice #+BEGIN_proof @@ -1834,11 +1978,18 @@ Soit $f\colon x \mapsto \sum_{n=1}^{+\i} \frac{1}{n} \sin\left(\frac{x}{4^n}\rig Si on prend un $x = 4^n \pi$, on est très proche de $0$. - Pourquoi est-on forcément positif ?? La fonction est impaire… - 2. Simple, cf ce qui précède. + Il me semble que $\inf \{f(t),\, t\geq x\}$ peut prendre des + valeurs arbitrairement petites (donc négative), ce qui contredit + l'énoncé d'origine. On peut montrer + + En effet, par segments emboîtés, on peut trouver $x\in [0, 2\pi]$ + tel que $\forall n\in\N,\, \sin( 4^n x)\leq 0$. + 2. Prendre $x_0 = \frac{4\pi}{3}$ et $t_N = 4^N x_0$. + 3. Simple, cf ce qui précède. #+END_proof +# ID:8849 #+begin_exercice [ENS L 2025 # 113] Soit $(\lambda_n)$ une suite de réels $\gt 0$ telle que $\forall n \in \N, \ 2\lambda_n \leq \lambda_{n+1} \leq 3\lambda_n$. Montrer que : @@ -1849,61 +2000,102 @@ $$\forall \alpha \gt 0, \ \exists (c_1, c_2) \in (\R^{+*})^2, \ \frac{c_2}{(1-t)^{\alpha}}$$. #+end_exercice #+BEGIN_proof -Calculons $\sum_{n\geq 1} 2^{n\a} t^{2^n}$ +La valeur de la somme est donnée (à une constant près) par son plus grand terme. + + $\frac{u_{n+1}}{u_n} = C^{\a} t^{\la_{n+1} - \la_n}$. Si à un instant ça vaut $1$ : + + l'instant d'après ça vaut $\frac{1}{2}$, donc la partie à droite est négligeable. + + l'instant d'avant ça vaut $2$, donc la partie à gauche est négligeable. #+END_proof +# ID:8845 #+begin_exercice [ENS SR 2025 # 114] On pose : $\forall x \gt 0, \eta(x) = \sum_{n=0}^{+\i} \frac{(-1)^{n-1}}{n^x}$. 1. Montrer que $\eta$ est de classe $C^{\i}$ sur $]1, +\i[$. Étudier sa limite en $+\i$. 1. Montrer que $\eta$ est de classe $C^{\i}$ sur $]0, +\i[$. 1. Calculer $\eta(1)$. - 1. Montrer que : $\forall z \in \C, |e^z 1| \leq e^{|z|} 1$. + 1. Montrer que : $\forall z \in \C, |e^z - 1| \leq e^{|z|}- 1$. 1. Montrer que $\eta(z)$ est bien définie pour tout $z \in \C$ vérifiant $\Re z \gt 0$. #+end_exercice -#+begin_exercice [ENS P 2025 # 115] +#+begin_exercice [ENS P 2025 # 115] :todo: 1. Montrer que, pour tout $n \in \N$, il existe un unique $L_n \in \R[X]$ tel que $L_n(1) = 1$ et $(1 - X^2)L_n'' - 2XL_n' + n(n+1)L_n = 0$. - 1. Montrer que $\forall x \in [-1,1], \ \forall z \in ]-1,1[, \ \frac{1}{\sqrt{1-2xz+z^2}} = \sum_{n=1}^{+\i} L_n(x)z^n$. + 1. Montrer que $\forall x \in [-1,1], \ \forall z \in \interval]{-1, 1}[, \ \frac{1}{\sqrt{1-2xz+z^2}} = \sum_{n=1}^{+\i} L_n(x)z^n$. #+end_exercice +#+BEGIN_proof :todo: + 1. On considère l'application $\Phi\colon P\mapsto \dots$. Elle + envoie $\R_n[X]$ dans $\R_{n-1}[X]$, et de plus, si $\deg P\lt n$, + alors $\deg \Phi(P) = \deg P$. Cela donne le rang de la matrice. +#+END_proof +# ID:8869 #+begin_exercice [ENS PLSR 2025 # 116] -Soient $f,g\in\mc C^0([0,1],\R)$ telles que f(1)=g(1)=1 et, pour tout $x\in[0,1[,|f(x)|\lt 1$. On suppose qu'il existe C\gt 0 et $M\in\N^*$ tels que $1-f(1-x)\underset{x\ra 0^+}{\sim}Cx^{1/M}$. -Pour $n \in \N$, on pose $u_n = \int_0^1 g(x) f(x)^n dx$. +Soient $f,g\in\mc C^0([0,1],\R)$ telles que $f(1)=g(1)=1$ et, pour +tout $x\in[0,1[,|f(x)|\lt 1$. On suppose qu'il existe $C\gt 0$ et +$M\in\N^*$ tels que $1-f(1-x)\underset{x\ra 0^+}{\sim}Cx^{1/M}$. Pour +$n \in \N$, on pose $u_n = \int_0^1 g(x) f(x)^n dx$. 1. Déterminer un équivalent de $u_n$. - 1. Montrer l'existence de C' tel que : $\forall n \in \N^*, \left| \frac{u_{n+1}}{u_n} 1 \right| \leq \frac{C'}{n}$. + 1. Montrer l'existence de $C'$ tel que : $\forall n \in \N^*, \left| + \frac{u_{n+1}}{u_n}- 1 \right| \leq \frac{C'}{n}$. #+end_exercice +#+BEGIN_proof + 1. Classique. + 2. On a $u_{n+1} - u_n = \int_{0}^1 g(x) (f(x) - 1) f(x)^n\dx$. +#+END_proof + +# ID:8862 #+begin_exercice [ENS SR 2025 # 117] Soit $f\colon x \mapsto \int_0^{+\i} \cos\left(\frac{t^3}{3} + tx\right) dt$. 1. Montrer la définition de $f$ sur $\R^+$ 1. Soit $x \geq 0$. Montrer que Re $\left[ \int_0^{+\i} \exp \left( i \left( \frac{(t+i\eps)^3}{3} + (t+i\eps)x \right) \right) dt \right] \xrightarrow[\eps \ra 0^+]{} f(x)$. #+end_exercice +#+BEGIN_proof + 1. Forcer une IPP : + $\int_1^{+\i} \cos(\phi(t)) \dt = \left[ \frac{\sin(\phi(t))}{t^2+x} \right]_1^{+\i} + \int_1^{+\i} \frac{2t}{(t^2+x)^2} \sin(\phi(t)) \dt$ + 2. Couper entre $0, 1$. Sur $[1,+\i]$ faire la même IPP. +#+END_proof + +# ID: 7504 #+BEGIN_exercice [ENS SR 2025 # 118] -On note $E$ l'ensemble des fonctions continues et de carré intégrable de $\R^{+*}$ -dans $C$. - - a. On convient que -$$\sqrt{+\i}=+\i$$ -. Pour $f$ continue de $\R^{+*}$ dans $\C$, montrer que -$$\sqrt{\int_0^{+\i}|f|^2}=\sup\left\{\int_0^{+\i}|fg|\;;\;g\in E\;\mathrm{tel}\;\mathrm{que}\;\int_0^{+\i}|g|^2=1\right\}$$. - - 1. Soit $f \in E$. Montrer que $\Phi : x \in \R^{+*} \mapsto \int_0^{+\i} \frac{f(t)}{t+x} dt$ appartient à $E$. +On note $E$ l'ensemble des fonctions continues et de carré intégrable +de $\R_+^{*}$ dans $\C$. + 1. On convient que $\sqrt{+\i}=+\i$. Pour $f$ continue de $\R_+^{*}$ + dans $\C$, montrer que + $$\sqrt{\int_0^{+\i}|f|^2}=\sup\left\{\int_0^{+\i}|fg|\;;\;g\in E\;\mathrm{tel}\;\mathrm{que}\;\int_0^{+\i}|g|^2=1\right\}$$. + 2. Soit $f \in E$. Montrer que $\Phi\colon x \in \R_+^{*} \mapsto + \int_0^{+\i} \frac{f(t)}{t+x} \dt$ appartient à $E$. #+END_exercice +#+BEGIN_proof + 1. Si c'est fini, c'est Cauchy-Schwarz. + + Si c'est infini, alors on peut prendre des Cauchy-Schwarz à des tronqués de $f$. + 2. Continuité : Pour tout segment $[a, b] \subset \R_+^*$, on a pour $x \in [a, b]$, $|\frac{f(t)}{t+x}| \leq \frac{|f(t)|}{t+a}$. Or $t \mapsto \frac{1}{t+a} \in L^2(\R_+^*)$, donc par Cauchy-Schwarz, $t \mapsto \frac{|f(t)|}{t+a}$ est intégrable. Le théorème de continuité des intégrales à paramètres assure que $\Phi$ est continue. + + Le caractère de carré intégrable est super dur, il faut faire du Cauchy-Schwarz en écrivant $\frac{|f(t)|}{t+x} = \left( |f(t)| \frac{(t/x)^{1/4}}{\sqrt{t+x}} \right) \left( \frac{(x/t)^{1/4}}{\sqrt{t+x}} \right)$ +#+END_proof -#+begin_exercice [ENS P 2025 # 119] -Soient $K \in \mc C^0([0,1]^2,\R)$ telle que $||K||_{\i} \lt 1$ et $f \in \mc C^0([0,1],\R)$. Étudier l'existence et l'unicité de $g \in \mc C^0([0,1],\R)$ telle que $\forall x \in [0,1], \ g(x) \int_{\R}^1 K(x,t)g(t) \ dt = f(x)$. + +# Relier à 4304, 4305 +#+begin_exercice [ENS P 2025 # 119] :todo: +Soient $K \in \mc C^0([0,1]^2,\R)$ telle que $||K||_{\i} \lt 1$ et $f \in \mc C^0([0,1],\R)$. Étudier l'existence et l'unicité de $g \in \mc C^0([0,1],\R)$ telle que $\forall x \in [0,1], \ g(x) - \int_{\R}^1 K(x,t)g(t) \ dt = f(x)$. #+end_exercice +#+BEGIN_proof + +#+END_proof + #+begin_exercice [ENS L 2025 # 120] Soient $\alpha, \theta \in ]0,1[$. Pour $f:[1,+\i[ \ra [0,1]$ continue, on pose $\|f\|_{\alpha}=\sup_{s \ra \i} s^{\alpha}|f(s)|$ et $F_{\alpha}=\{f \in \mc C^0([1,+\i[,[0,1]),\,\|f\|_{\alpha}\lt +\i\}$. 1. Pour $f \in F_{\alpha}$, on pose $T(f): s \geq 1 \mapsto 1 - \left(1 - \frac{1}{s}\right)^{\theta} + \theta(s-1)^{\theta} \int_{-\i}^{+\i} (s+t-1)^{-\theta-1} f(t) dt$. -Montrer que $T$ est une application lipschitzienne de $F_{\alpha}$ dans $F_{\alpha}$ (pour $\| \cdot \|_{\alpha}$).- b) On admet que, pour tout $\alpha \in ]0, 1-\theta[$, $T$ possède un unique point fixe $f_{\alpha} \in F_{\alpha}$. Montrer que $f_{\alpha}$ ne dépend pas de $\alpha$ ; on le note $f_0$. Montrer que $\int_t^{+\i} t^{-\theta} f_0(t) dt = +\i$. +Montrer que $T$ est une application lipschitzienne de $F_{\alpha}$ dans $F_{\alpha}$ (pour $\| \cdot \|_{\alpha}$). + 2. On admet que, pour tout $\alpha \in ]0, 1-\theta[$, $T$ possède un unique point fixe $f_{\alpha} \in F_{\alpha}$. Montrer que $f_{\alpha}$ ne dépend pas de $\alpha$ ; on le note $f_0$. Montrer que $\int_t^{+\i} t^{-\theta} f_0(t) dt = +\i$. #+end_exercice +# ID:8846 #+begin_exercice [ENS PLSR 2025 # 121] 1. Expliciter le terme général d'une suite $(a_n)_{n\geq 0}$ vérifiant la relation de récurrence $na_{n+1} = (n+1)a_n$ pour @@ -1912,12 +2104,17 @@ Montrer que $T$ est une application lipschitzienne de $F_{\alpha}$ dans $F_{\alp #+end_exercice #+begin_exercice [ENS PLSR 2025 # 122] -Résoudre $x^2y'' + xy' + (x^2 1/4)y = 0$ sur $]0, 1[$. +Résoudre $x^2y'' + xy' + (x^2 - 1/4)y = 0$ sur $]0, 1[$. #+end_exercice +# ID: nil #+begin_exercice [ENS P 2025 # 123] -Soient $(a, b) \in \R^2$ avec $a \lt b, \psi \in \mc C^2([a, b], \R^{+*})$ croissante. Soit $y \in \mc C^2([a, b], \R)$ non nulle et vérifiant $y'' + \psi(x)y = 0$. Montrer que les points où |y| admet un extremum local forment une suite finie $(a_1, \ldots, a_n)$ (éventuellement vide) et que la suite des valeurs $(|y(a_1)|, \ldots, |y(a_n)|)$ est décroissante. +Soient $(a, b) \in \R^2$ avec $a \lt b, \psi \in \mc C^2([a, b], \R^{+*})$ croissante. Soit $y \in \mc C^2([a, b], \R)$ non nulle et vérifiant $y'' + \psi(x)y = 0$. Montrer que les points où $|y|$ admet un extremum local forment une suite finie $(a_1, \ldots, a_n)$ (éventuellement vide) et que la suite des valeurs $(|y(a_1)|, \ldots, |y(a_n)|)$ est décroissante. #+end_exercice +#+BEGIN_proof +C'est faux : pour $\psi(x) = 1$, $y$ est la fonction cosinus, et entre deux zéros, il existe un extremum local, donc machin n'est pas décroissante. +#+END_proof + #+begin_exercice [ENS PLSR 2025 # 124] Soit $f \in \mc C^2(\R, \C)$. @@ -1929,6 +2126,7 @@ On pose $\partial_P f = \sum_{k=0}^{\i} a_k f^{(k)}$. Donner une condition néce $$\forall (x,y,z) \in \mc C^1(\R,\C)^3, \quad \left\{ \begin{array}{l} x' + ax + by + cz \xrightarrow{+\i} 0 \\ y' + bx + cy + az \xrightarrow{+\i} 0 \\ z' + cx + ay + bz \xrightarrow{+\i} 0 \end{array} \right. \Longrightarrow \left\{ \begin{array}{l} x \xrightarrow{+\i} 0 \\ y \xrightarrow{+\i} 0 \\ z \xrightarrow{+\i} 0 \end{array}\right.$$ #+end_exercice +# ID:nil # Simple #+begin_exercice [ENS SR 2025 # 125] Soient $I$ un intervalle de $\R$ et $A:I\ra\M_2(\R)$ continue. On regarde l'équation (1): $X'(t) = A(t) X(t)$. 1. Décrire l'ensemble des solutions de (1). @@ -1936,6 +2134,7 @@ Soient $I$ un intervalle de $\R$ et $A:I\ra\M_2(\R)$ continue. On regarde l'équ Trouver une condition sur $D$ pour que les solutions de (1) aient une limite quand $t \ra +\i$. #+end_exercice +# ID:nil # Simple #+begin_exercice [ENS P 2025 # 126] Soit $n \ge 2$. Soit $A : \R^+ \ra \M_n(\R)$ continue. On considère les solutions de l'équation différentielle $(): x'(t) = A(t)x(t)$. 1. On suppose qu'il existe $P \in GL_n(\R)$ et $D : \R^+ \ra \M_n(\R)$ continue et à valeurs dans l'ensemble des matrices diagonales à coefficients dans $]-\i,-1]$ telles que, pour tout $t$, $A(t) = P D(t) P^{-1}$. Les solutions de () ont-elles toutes pour limite 0 en $+\i$ ? @@ -1961,22 +2160,31 @@ Soit $\mu \in \R^+$. On considère $(E_\mu): y'' \mu(1-y^2)y' + y = 0$. Calculer alors $\omega_1$ et donner une expression explicite de $x_0$ et $x_1$ en fonction de quelques constantes inconnues. #+end_exercice +# ID:180 # Classique #+begin_exercice [ENS L 2025 # 129] -Soit $A$ une application continue de $\R$ dans $\M_n(\C)$ et $X$ une application de classe $C^1$ de $\R$ dans $\M_n(\C)$ telles que, pour tout $t \in \R$, X'(t) = A(t)X(t) X(t)A(t). Montrer que, pour tout $t \in \R$, X(t) est semblable à X(0). +Soit $A$ une application continue de $\R$ dans $\M_n(\C)$ et $X$ une application de classe $C^1$ de $\R$ dans $\M_n(\C)$ telles que, pour tout $t \in \R$, $X'(t) = A(t)X(t)-X(t)A(t)$. Montrer que, pour tout $t \in \R$, $X(t)$ est semblable à $X(0)$. #+end_exercice +# ID:8238 #+begin_exercice [ENS SR 2025 # 130] -Soit $f\colon \R^2 \ra \R$ telle que $f(x,y) = \frac{e^x e^y}{x y}$ si $x \neq y$ et $f(x,x) = e^x$. Montrer que $f$ est de classe $\mc C^{\i}$. +Soit $f\colon \R^2 \ra \R$ telle que $f(x,y) = \frac{e^x -e^y}{x -y}$ si $x \neq y$ et $f(x,x) = e^x$. Montrer que $f$ est de classe $\mc C^{\i}$. #+end_exercice +# ID:8847 #+begin_exercice [ENS P 2025 # 131] Soient $d\in\N^*$ et $f\colon\R^d\ra\R$ de classe $\mc C^2$. Soit $L\geq\ell\gt 0$ des réels. On suppose qu'en tout point de $\R^d$ la hessienne de $f$ a son spectre inclus dans $[\ell,L]$. Soit $\tau\in ]0,2/L[$ ainsi qu'une suite $u$ à termes dans $\R^d$ vérifiant la relation de récurrence $\forall n\in\N,\ u_{n+1}=u_n-\tau\nabla f(u_n)$. Montrer que $u$ converge. #+end_exercice +#+BEGIN_proof +En écrivant $g(t) = u_n - t \nable f (u_n)$, et en écrivant $f(g(1)) - f(g(0))$ comme une intégrale double on trouve que c'est décroissant, donc $f(u_n)$ converge, mais $f$ est propre. Si $u_n$ admet une valeur d'adhérence, alors c'est forcément un point critique (sinon la valeur de $f$ décroît strictement), et il n'y a qu'au plus un point critique. +#+END_proof +# ID:nil #+begin_exercice [ENS PLSR 2025 # 132] Soient $d \in \N^*$ et $f : \R^d \ra \R$ de classe $\mc C^1$. On suppose que $f$ tend vers $+\i$ en $\i$, que $\nabla f$ est lipschitzienne et que les points critiques de $f$ sont isolés dans $\R^d$. Montrer qu'il existe un réel $\tau \gt 0$ tel que, quel que soit le choix de $a \in \R^d$, la suite définie par $x_0 = a$ et $\forall n \in \N, \ x_{n+1} = x_n - \tau \nabla f(x_n)$ soit convergente. On commencera par le cas où $d=1$ et $f\colon x \mapsto \frac{x^2}{2}$. #+end_exercice + +# ID:8850 #+begin_exercice [ENS L 2025 # 133] Soit $G$ un sous-groupe fermé de $\mathrm{GL}_n(\R)$. @@ -1986,14 +2194,25 @@ On pose $L = \{A \in \M_n(\R) ; \forall t \in \R, e^{tA} \in G\}$. 1. Montrer que $\forall (A, B) \in L^2$, $AB - BA \in L$. 1. Que peut-on dire de $L$ pour $G = \mathrm{SL}_n(\R)$ ? #+end_exercice +#+BEGIN_proof + 1. On sait que $\big(e^{tA/k}e^{tB/k}\big)^k$ est dans le groupe. Il + s'agit de justifier que cette quantité tend vers $e^{t(A+B)}$. + Ça peut se faire par un DL, en utilisant $(1 + A/n + O(\frac{1}{n^2}))^n \ra e^A$. + 2. On considère $e^{tA} e^{tB} e^{-tA} e^{-tB}$, on met des $\frac{1}{k}$ et des ${\cdot}^k$.. + 3. +#+END_proof + + +# ID:8851 #+begin_exercice [ENS PLSR 2025 # 134] Soit $n \ge 2$ un entier. Une application $f$ de classe $C^2$ définie sur un ouvert $O$ de $\R^n$, à valeurs dans $\R^n$ vérifie la propriété $\mc{P}$ si, pour tout $x \in O$, $df_x$ est composée d'une homothétie et d'une isométrie vectorielle. 1. On suppose que $n=2$ et que $f$ vérifie $\mc{P}$. On note $f=(f_1,f_2)$. Montrer que $f_1$ et $f_2$ sont harmoniques, c'est-à-dire que $\Delta f_1=0$ et $\Delta f_2=0$. - 1. Montrer que le résultat de la question a) est faux si $n \geq 3$. On pourra considérer l'application $f\colon x \in \R^n \setminus \{0\} \mapsto \frac{x}{\|x\|^2}$. + 2. Montrer que le résultat de la question précédente est faux si $n \geq 3$. On pourra considérer l'application $f\colon x \in \R^n \setminus \{0\} \mapsto \frac{x}{\|x\|^2}$. #+end_exercice #+BEGIN_proof - 1. + 1. Si $df_x = \la I_n$, $df(x) = \begin{pmatrix}\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y}\end{pmatrix}$ alors quand on dérive par rapport à $x$, on est une matrice antisymétrique, plus $\mu$ identité. $\frac{\partial^2 f_1}{\partial^2 x} = \frac{\partial^2 f_2}{\partial x\partial y}$ et $\frac{\partial^2 f_2}{\partial^2 x} = - \frac{\partial^2 f_1}{\partial x\partial y}$ + et même chose en dérivant par rapport à $y$ : $\frac{\partial^2 f_1}{\partial^2 y} = - \frac{\partial^2 f_2}{\partial x\partial y}$. 2. #+END_proof @@ -2012,7 +2231,6 @@ En général, on va faire un changement de coordonnées polaires : En général, la condition d'homogénéité donne $r\frac{\partial f}{\partial r} = \la f$ , donc avec le laplacien, on obtient $\la^2 f + \frac{\partial^2 f}{\partial \theta^2} = 0$. Donc à $r$ fixé, $f$ dépend de $\theta$ comme $\a \cos (\sqrt{\la}\theta + c)$, ce qui impose que $\la$ soit le carré d'un entier. #+END_proof - ** Géométrie # ID:8728 @@ -2026,10 +2244,10 @@ On sait paramétriser les triplets pythagoricien : $(a,b) = (m^2 - n^2, 2mn)$, d #+END_proof -#+begin_exercice [ENS P 2025 # 137] +#+begin_exercice [ENS P 2025 # 137] :todo: Soient $a$, $b$, $c$, $d$ dans $\R_+^*$. Quelle est l'aire maximale d'un quadrilatère dont les côtés successifs ont pour longueurs $a$, $b$, $c$, $d$ ? #+end_exercice -#+BEGIN_proof +#+BEGIN_proof :todo: Paramétrer par la distance entre le premier sommet et le troisième, $x$. Il faut que $|b-a|\leq x\leq a+b$. Puis pour trouver l'aire d'un triangle $a,b,x$, on pose la hauteur $h$, qui coupe $x$ en $x = u+v$, @@ -2045,10 +2263,21 @@ Toujours plus agréable via de la trigo… #+END_proof #+begin_exercice [ENS PLSR 2025 # 138] - 1. Quelle est l'aire maximale possible pour un rectangle de périmètre 1? - 1. On considère un entier $n \geq 3$ et une liste strictement croissante $(\theta_1, \dots, \theta_n)$ à termes dans $[0, 2\pi]$. Déterminer la valeur maximale possible pour le périmètre du polygone de sommets $e^{i\theta_1}, \dots, e^{i\theta_n}$ (dans cet ordre). - 1. Soit $z_1,\ldots,z_n$ des nombres complexes. On convient que $z_0=z_n$. On définit l'aire algé- -brique du polygone $z_1\cdots z_n$ comme $\frac{1}{2}\sum_{k=0}^{n-1}(\op{Re}(z_k)\op{Im}(z_{k+1})-\op{Im}(z_k)\op{Re}(z_{k+1}))$. On fixe un réel p\gt 0. Parmi les listes $(z_1,\ldots,z_n)\in\C^n$ telles que le périmètre de $z_1\cdots z_n$ soit égal à $p$, déterminer celles qui maximisent l'aire algébrique du polygone associé. + 1. Quelle est l'aire maximale possible pour un rectangle de périmètre 1 ? + 1. On considère un entier $n \geq 3$ et une liste strictement + croissante $(\theta_1, \dots, \theta_n)$ à termes dans $[0, + 2\pi]$. Déterminer la valeur maximale possible pour le périmètre + du polygone de sommets $e^{i\theta_1}, \dots, e^{i\theta_n}$ (dans + cet ordre). + 1. Soit $z_1,\ldots,z_n$ des nombres complexes. On convient que + $z_0=z_n$. On définit l'aire algébrique du polygone $z_1\cdots + z_n$ comme + $\frac{1}{2}\sum_{k=0}^{n-1}(\op{Re}(z_k)\op{Im}(z_{k+1})-\op{Im}(z_k)\op{Re}(z_{k+1}))$. + + On fixe un réel $p\gt 0$. Parmi les listes + $(z_1,\ldots,z_n)\in\C^n$ telles que le périmètre de $z_1\cdots + z_n$ soit égal à $p$, déterminer celles qui maximisent l'aire + algébrique du polygone associé. #+end_exercice ** Probabilités @@ -2116,7 +2345,7 @@ Soit $\mc{P}$ l'ensemble des nombres premiers et $\beta \gt 1$. Soit $(Y_p)_{p Montrer qu'il existe $C \gt0$ tel que pour tout $n \ge 1$ et tout $(a_{i,j})_{1 \le i,j \le n} \in \{\pm 1\}^{n^2}$, il existe $(x_i)_{1 \le i \le n}$ et $(y_i)_{1 \le i \le n}$ dans $\{\pm 1\}^n$ tels que $\sum_{1 \le i \le n} a_{i,j} x_i y_j \ge C n^{3/2}$. #+end_exercice -#+begin_exercice [ENS MP 2025 # 145] +#+begin_exercice [ENS 2025 # 145] :todo: Soient $\theta \in ]0,1[$ et $X$ une variable aléatoire à valeurs dans $\R^+$ telle que $\mathbf{P}(X\gt 0)\gt 0$. Montrer que $\mathbf{P}(X\geq\theta\,\mathbf{E}(X))\geq \frac{(1-\theta)^2\mathbf{E}(X)^2}{\mathbf{E}(X^2)}$. #+end_exercice @@ -2175,16 +2404,16 @@ autre premier pas ne le fait plus revenir en $0$. #+END_proof -#+begin_exercice [ENS P 2025 # 148] -Soient $p \in ]0, 1/2[$ et $(X_n)_{n \ge 1}$ une suite $i$.i.d. de variables aléatoires telle que $\mathbf{P}(X_n=1)=1-\mathbf{P}(X_n=-1)=p. \text{ Pour } n\in\N^*, \text{ on note } S_n=X_1+\cdots+X_n. \text{ Montrer } 1\text{'existence de } c,C_1,C_2\gt 0 \text{ tels que } \forall u\geq 0,\ C_1e^{-cu}\leq \mathbf{P}\left(\sup_{n\geq 1}S_n\geq u\right)\leq C_2e^{-cu}$. +#+begin_exercice [ENS P 2025 # 148] :todo: +Soient $p \in ]0, 1/2[$ et $(X_n)_{n \ge 1}$ une suite $i$.i.d. de variables aléatoires telle que $\mathbf{P}(X_n=1)=1-\mathbf{P}(X_n=-1)=p$. Pour $n\in\N^*$, on note $S_n=X_1+\cdots+X_n$. Montrer l'existence de $c,C_1,C_2\gt 0$ tels que $\forall u\geq 0,\ C_1e^{-cu}\leq \mathbf{P}\left(\sup_{n\geq 1}S_n\geq u\right)\leq C_2e^{-cu}$. #+end_exercice #+begin_exercice [ENS PLSR 2025 # 149] - 1. Soit $X$ une variable aléatoire réelle et $s$ \gt 0 tel que $\mathbf{E}(e^{sX})$ soit finie. Démontrer que $\forall a \gt 0$, $\mathbf{P}(X \ge a) \le e^{-sa} \mathbf{E}(e^{sX})$. + 1. Soit $X$ une variable aléatoire réelle et $s\gt 0$ tel que $\mathbf{E}(e^{sX})$ soit finie. Démontrer que $\forall a \gt 0$, $\mathbf{P}(X \ge a) \le e^{-sa} \mathbf{E}(e^{sX})$. 1. Soit $(X_i)_{i \ge 1}$ une suite de variable aléatoires $i$.i.d. à valeurs dans [0, 1]. -On pose $S_n = \sum_{i=1}^n X_i$. Démontrer que $\forall t \gt 0$, $\mathbf{P}(|S_n - \mathbf{E}(S_n)| \ge t) \le 2e^{-t^2/(2n)}$. + On pose $S_n = \sum_{i=1}^n X_i$. Démontrer que $\forall t \gt 0$, $\mathbf{P}(|S_n - \mathbf{E}(S_n)| \ge t) \le 2e^{-t^2/(2n)}$. #+end_exercice #+begin_exercice [ENS PLSR 2025 # 150] @@ -2241,27 +2470,53 @@ Soient $(X_k)_{k \in \N^*}$ une suite de variables de Rademacher indépendantes #+end_exercice #+begin_exercice [ENS P 2025 # 154] -On considère $n$ variables aléatoires de Rademacher indépendantes $(\eps_i)_{1\leq i\leq n}$. Montrer que, pour tout réel $p \gt 0$, il existe $(c_p, C_p) \in (\R^{+*})^2$ indépendant de $n \in \N^*$ tel que, +On considère $n$ variables aléatoires de Rademacher indépendantes +$(\eps_i)_{1\leq i\leq n}$. Montrer que, pour tout réel $p \gt 0$, il +existe $(c_p, C_p) \in (\R^{+*})^2$ indépendant de $n \in \N^*$ tel +que, pour tout $(z_1, \ldots z_n) \in \C^n$, $$c_p \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}} \leq \left(\mathbf{E} \left|\sum_{i=1}^n \eps_i z_i\right|^p\right)^{\frac{1}{p}} \leq C_p \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}}.$$ #+end_exercice +# ID:8866 #+begin_exercice [ENS L 2025 # 155] Soit $(X_n)_{n\geq 0}$ une suite de variables aléatoires indépendantes à valeurs dans $\Z$ telles que $\forall n \in \N, \ \forall k \in \N, \ \mathbf{P}(X_n = k) = \mathbf{P}(X_n = -k) = ce^{-|k|}$ où $c$ est à déterminer. Déterminer la loi du rayon de convergence de la série entière aléatoire $\sum X_n z^n$. #+end_exercice +#+BEGIN_proof +On a presque sûrement $X_n \not \ra 0$, donc $R\leq 1$. -#+begin_exercice [ENS P 2025 # 156] -Soit $(p_n)_{n\geq 1}$ une suite d'éléments de [0,1]. Pour $n\in\N^*$, on note $G_n$ le graphe aléatoire $G_{n,p_n}$ d'Erdös-Renyi, c'est-à-dire un graphe aléatoire de sommets [1,n] et une famille $(X_{\{i,j\}})_{\{i,j\}\in\mc{P}_2([1,n])}$ de variables de Bernoulli $i$.i.d. de paramètre $p_n$, avec $X_{\{i,j\}}=$ 1 si et seulement s'il existe une arête reliant $i$ et $j$. On note $I_n$ le nombre de sommets isolés $de G_n$. - 1. Soit $\eps \in ]0,1[$. On suppose que, pour tout $n \in \N^*$, $p_n \geq (1+\eps)\frac{\ln(n)}{n}$. Montrer que $\mathbf{P}(I_n \geq 1) \xrightarrow[n \ra +\i]{} 0$. - 1. Soit $\eps \in ]0,1[$. On suppose que, pour tout $n \in \N^*$, $p_n \leq (1-\eps)\frac{\ln(n)}{n}$. Montrer que $\mathbf{P}(I_n \geq 1) \underset{n \ra +\i}{\longrightarrow} 1$. +Soit $c\in \interval]{0, 1}[$, $\P(|c^n X_n|\geq 1)\simeq e^{-1/c^n}$, donc $\P\big(\exists \i n \mid |c^n X_n|\geq 1\big) = 1$, donc le rayon vaut $1$. +#+END_proof + + +#+begin_exercice [ENS P 2025 # 156] :todo: +Soit $(p_n)_{n\geq 1}$ une suite d'éléments de $[0,1]$. Pour +$n\in\N^*$, on note $G_n$ le graphe aléatoire $G_{n,p_n}$ +d'Erdös-Renyi, c'est-à-dire un graphe aléatoire de sommets $\db{1,n}$ +et une famille $(X_{\{i,j\}})_{\{i,j\}\in\mc{P}_2([1,n])}$ de +variables de Bernoulli $i$.i.d. de paramètre $p_n$, avec +$X_{\{i,j\}}=$ 1 si et seulement s'il existe une arête reliant $i$ et +$j$. On note $I_n$ le nombre de sommets isolés de $G_n$. + 1. Soit $\eps \in ]0,1[$. On suppose que, pour tout $n \in \N^*$, + $p_n \geq (1+\eps)\frac{\ln(n)}{n}$. Montrer que $\mathbf{P}(I_n + \geq 1) \xrightarrow[n \ra +\i]{} 0$. + 1. Soit $\eps \in ]0,1[$. On suppose que, pour tout $n \in \N^*$, + $p_n \leq (1-\eps)\frac{\ln(n)}{n}$. Montrer que $\mathbf{P}(I_n + \geq 1) \underset{n \ra +\i}{\longrightarrow} 1$. #+end_exercice +# ID:8867 #+BEGIN_exercice [ENS L 2025 # 157] -Montrer qu'il existe un réel $c\gt 0$ vérifiant la condition suivante : quel que soit $n \in \N^*$, quelle que soit $S$ partie non vide de $\mathbb{U}_n$, il existe un entier naturel $p \leq \frac{cn}{|S|}$ ainsi qu'une $p$-liste $(z_1,\ldots,z_p)$ d'éléments de $\mathbb{U}_n$ telle que $\left|\bigcup_{k=1}^p z_k S\right|\geq \frac{n}{2}$. +Montrer qu'il existe un réel $c\gt 0$ vérifiant la condition +suivante : quel que soit $n \in \N^*$, quelle que soit $S$ partie non +vide de $\mathbb{U}_n$, il existe un entier naturel $p \leq +\frac{cn}{|S|}$ ainsi qu'une $p$-liste $(z_1,\ldots,z_p)$ d'éléments +de $\mathbb{U}_n$ telle que $\left|\bigcup_{k=1}^p z_k S\right|\geq +\frac{n}{2}$. #+END_exercice #+BEGIN_proof -Méthode probabiliste. +Méthode probabiliste : $X = \Big|\bigcup z_k S \Big| = \sum_{x\in\m U_n}\sum_{s\in S} \m 1_{z_ks = x}$, donc $\E(X) = |S| p$, et on veut $|S| p\geq \frac{n}{2}$. #+END_proof # ID:8723 @@ -2294,13 +2549,29 @@ Soit $n \ge 3$. Une alpiniste dispose de $n$ lieux possibles pour planter sa ten 1. On fixe un $k_n$ maximisant $p_{n,k}$ lorsque $k$ parcourt $[0, n-1]$. Étudier le comportement asymptotique de $k_n$ quand $n$ tend vers $+\i$. #+end_exercice +# ID:8865 #+begin_exercice [ENS L 2025 # 160] -Soit $(X_n)_{n\geq 1}$ une suite $i$.i.d. de variables aléatoires réelles discrètes. Pour $t\in\R$ et $n \in \N^*$, on considère la variable aléatoire $f_n(t) = \frac{1}{n} \left| \{k \in [1, n], X_k \leq t\} \right|$. Montrer qu'il existe une fonction $f\colon \R \ra \R$ telle que $\mathbf{P}\left(\sup_{t \in \R} |f_n(t) - f(t)| \gt \eps\right) \underset{n \ra +\i}{\longrightarrow} 0$ pour tout réel $\eps \gt 0$. +Soit $(X_n)_{n\geq 1}$ une suite $i$.i.d. de variables aléatoires +réelles discrètes. Pour $t\in\R$ et $n \in \N^*$, on considère la +variable aléatoire $f_n(t) = \frac{1}{n} \left| \{k \in [1, n], X_k +\leq t\} \right|$. Montrer qu'il existe une fonction $f\colon \R \ra +\R$ telle que $\mathbf{P}\left(\sup_{t \in \R} |f_n(t) - f(t)| \gt +\eps\right) \underset{n \ra +\i}{\longrightarrow} 0$ pour tout réel +$\eps \gt 0$. #+end_exercice +#+BEGIN_proof +Si on fixe un $t$, on a $f_n(t)\tend{n\ra +\i} \P(X\leq t)$, d'après +la loi faible sur une suite de Bernoulli. + +Si $X$ est à support fini, il suffit de montrer la convergence en chaque $t_i$. + +Sinon, on écrit $X = X_f \m 1_{A_f} + Y_f 1_{\ol{A_f}}$, où $\P(A_f)$ est proche de $1$, alors $\P\big(|f_n(t) - f_{n,f}(t)|\geq \eps\big)\leq \P\big(A_f \text{ se produit plein de fois}\big)$. +#+END_proof + #+begin_exercice [ENS L 2025 # 161] Pour deux variables aléatoires réelles bornées $X$ et $Y$, sur des -espaces probabilisés a *priori* distincts, on note $X \leq_c Y$ pour +espaces probabilisés a priori distincts, on note $X \leq_c Y$ pour signifier que $\mathbf{E}(f(X)) \leq \mathbf{E}(f(Y))$ pour toute fonction convexe $f\colon \R \ra \R$. On se donne, sur un espace probabilisé, deux suites $(M, X_1, X_2, \dots)$ et $(N, Y_1, Y_2, @@ -2313,25 +2584,43 @@ conditions suivantes: On pose $S = \sum_{k=1}^M X_k$ et $T = \sum_{k=1}^N Y_k$. Montrer que $S \preccurlyeq_c T$. #+end_exercice +# ID:8864 #+begin_exercice [ENS L 2025 # 162] -Soient $E$ une partie bornée et au plus dénombrable de $\R^+$, et $\mc{L}$ et $\mc{L}'$ deux lois de probabilité sur $E$. Déterminer, en fonction de ces lois, la plus petite constante $K_{\mc{L},\mc{L}'}$ telle que, pour tout couple (X,Y) de variables aléatoires réelles à valeurs dans $E$ telles que $X \sim \mc{L}$ et $Y \sim \mc{L}'$, on ait l'inégalité $\mathbf{E}(XY) \leq K_{\mc{L},\mc{L}'}$. +Soient $E$ une partie bornée et au plus dénombrable de $\R^+$, et +$\mc{L}$ et $\mc{L}'$ deux lois de probabilité sur $E$. Déterminer, en +fonction de ces lois, la plus petite constante $K_{\mc{L},\mc{L}'}$ +telle que, pour tout couple $(X,Y)$ de variables aléatoires réelles à +valeurs dans $E$ telles que $X \sim \mc{L}$ et $Y \sim \mc{L}'$, on +ait l'inégalité $\mathbf{E}(XY) \leq K_{\mc{L},\mc{L}'}$. #+end_exercice +#+BEGIN_proof +C'est du réordonnement. +#+END_proof +# ID:8863 #+begin_exercice [ENS SR 2025 # 163] On munit $\R^n$ de sa structure euclidienne canonique. Soit $X=(X_1,\ldots,X_n)^T$ un vecteur aléatoire tel que $\mathbf{E}\left(\|X\|^2\right)\lt +\i$. On note $C(X)=\left(\op{Cov}(X_i,X_j)\right)_{1\leq i,j\leq n}$ la matrice de covariance. - 1. Que dire de C(X) si les $X_i$ sont indépendantes? - 1. Soient $v \in \R^n$ et $Y = \langle v, X \rangle$. Exprimer $\mathbf{V}(Y)$ en fonction de C(X). - 1. On suppose les $X_i$ centrées. Soient $A \in \M_n(\R)$ et $Z$ = AX. Exprimer $\mathbf{E}(\|Z\|^2)$ en fonction de C(X). - 1. Caractériser les $A \in \M_n(\R)$ pour lesquelles il existe un vecteur aléatoire $X$ tel que $A$ = C(X). - 1. Soit $H$ un hyperplan de $\R^n$. -Montrer que $P(X \in H) = 1$ si et seulement si $H^{\perp} \subset \text{Ker}(C(X))$. + 1. Que dire de $C(X)$ si les $X_i$ sont indépendantes? + 2. Soient $v \in \R^n$ et $Y = \langle v, X \rangle$. Exprimer $\mathbf{V}(Y)$ en fonction de $C(X)$. + 3. On suppose les $X_i$ centrées. Soient $A \in \M_n(\R)$ et $Z$ = AX. Exprimer $\mathbf{E}(\|Z\|^2)$ en fonction de $C(X)$. + 4. Caractériser les $A \in \M_n(\R)$ pour lesquelles il existe un vecteur aléatoire $X$ tel que $A = C(X)$. + 5. Soit $H$ un hyperplan de $\R^n$. Montrer que $P(X \in H) = 1$ si et seulement si $H^{\perp} \subset \op{Ker}(C(X))$. #+end_exercice +#+BEGIN_proof + 1. Elle est diagonale. + 2. $\V(Y) = v^T C(X) v$ + 3. $Z = \big(\langle L_i, X\rangle\big)$, où $L_i$ sont les lignes donc $\E\big(\lN Z\rN^2\big)$ est la somme des variances de ces choses. + 4. $C(X)$ est symétrique, positive. Réciproquement, $A = P^T D P$. + Pour $D$ on prend une somme de variables indépendantes, puis pour + $A$ on fait le changement de base. +#+END_proof + #+begin_exercice [ENS SR 2025 # 164] Soit $\alpha \gt 0$. On considère l'équation différentielle (): $(y' = -x, x' = \alpha^2 y)$ avec $(x,y):\R\ra\R^2$. 1. Si $(x_0, y_0) \in \R^2$ est fixé, justifier l'existence et l'unicité d'une solution de () vérifiant $x(0) = x_0$ et $y(0) = y_0$. Pour cette solution, on pose $I(t) = y^2(t)$ et $J(t) = \alpha^2 x^2(t)$. - 1. Montrer que les applications $T \mapsto \frac{1}{T} \int_0^T I(t) dt$ et $T \mapsto \frac{1}{T} \int_0^T J(t) dt$ admettent une limite finie en $+\i$. - 1. Soit $N \in \N^*$. On considère deux variables aléatoires $x_0, y_0$ indépendantes à valeurs dans + 2. Montrer que les applications $T \mapsto \frac{1}{T} \int_0^T I(t) dt$ et $T \mapsto \frac{1}{T} \int_0^T J(t) dt$ admettent une limite finie en $+\i$. + 3. Soit $N \in \N^*$. On considère deux variables aléatoires $x_0, y_0$ indépendantes à valeurs dans $\frac{1}{N}\Z$ telles que, pour tout $k \in \Z$, $\mathbf{P}\left(x_0 = \frac{k}{N}\right) = \mathbf{P}\left(y_0 = \frac{k}{N}\right) = \gamma_N \exp\left(-(k/N)^2\right)$. a) Justifier l'existence de $\gamma_N \in \R^+$ pour lequel ces conditions définissent la loi des deux variables aléatoires. b) On fixe $t$ et on considère, pour $N \in \N^*$, la variable aléatoire $f_N(t) = I(t) + J(t)$ (les fonctions $I$ et $J$ sont associées aux variables aléatoires $x_0$ et $y_0$). Montrer que $\mathbf{E}\left(e^{-f_N(t)}\right)$ possède une limite quand $N \ra +\i$. @@ -3517,12 +3806,20 @@ Les espaces $\R^p$ sont munis de leurs normes euclidiennes canoniques. Soient $d 1. Soient $p_0, \ldots, p_n \in \R^d$ tels que $||p_i p_j||^2 \in \Q$ pour tous $i, j \in [0, n]$. Montrer que $(p_0,\ldots,p_n)$ se plonge isométriquement dans $\Q^{4d}$. On admettra que tout entier naturel est somme de quatre carrés d'entiers. #+end_exercice +# ID:8875 #+BEGIN_exercice [X MP 2025 # 293] 1. Soit $A \in \mc{S}_n(\R)$. Montrer que $A$ est définie positive si et seulement si, pour tout $k \in [1, n], \det((a_{i,j})_{1 \le i, j \le k}) \gt 0$. 1. On pose $A_k = (t^{|i-j|})_{1 \le i,j \le k}$ où $t \in \R^{+*}$. Calculer $\det A_k$. 1. On pose $A = \left(\frac{1}{1+|i-j|}\right)_{1 \le i,j \le n}$. Démontrer que $A$ est symétrique définie positive. #+END_exercice +#+BEGIN_proof + 1. Classique : d'un côté les sous matrice carrés sont positives. + + De l'autre, par récurrence. See 6169 + 2. C'est un calcul. + 3. $A$ est l'intégrale des $A_k$. +#+END_proof #+begin_exercice [X MP 2025 # 294] @@ -3673,24 +3970,26 @@ Soient $\alpha \in \R^{+*}$ et $\beta = 1/\alpha$. Soit $(z_n)_{n \geq 0}$ la su #+END_proof +# ID:8874 #+BEGIN_exercice [X MP 2025 # 306] -Pour $n \in \N$, on pose $u_n = |\{(p,q) \in \N^2, ,\, p^2 + q^2 = n\}|$. +Pour $n \in \N$, on pose $u_n = |\{(p,q) \in \N^2 \mid p^2 + q^2 = n\}|$. 1. Déterminer la limite de la suite de terme général $\frac{1}{n}\sum u_k$. 1. Étudier la nature de la suite $(u_n)$. 1. Montrer que $(u_n)$ n'est pas bornée. #+END_exercice #+BEGIN_proof 1. Nombres de points entiers dans une boule. - 2. Elle diverge, car elle vaut $0$ souvent. + 2. Elle diverge, car elle vaut $0$ souvent, ou via Cesàro. 3. Si on sait que tout nombre premier $4k+1$ l'est, en prenant $p_1^2\dots p_k^2$ on a plein de façons de l'écrire. - Si on sait que si on est divisible par un nombre premier $4k+3$ et pas son carré, et que $\sum \frac{1}{p}$ diverge, pour les $4k+3$, on obtient que la proportion d'entiers pour lesquels c'est $0$ est $1$, je pense, ce qui suffit. + Si on sait que si on est divisible par un nombre premier $4k+3$ et + pas son carré, et que $\sum \frac{1}{p}$ diverge, pour les $4k+3$, + on obtient que la proportion d'entiers pour lesquels c'est $0$ est + $1$, je pense, ce qui suffit. - Si on ne sait pas ça. Il nous reste à regarder des $2^n \cdots 5^m$ : On a $2 = 1 + 1$ et $5 = 4+1$, $25 = 25 + 0 = 16 + 9$ a deux façons de l'écrire, $125 = 100 + 25 = 121 + 4$, n'a que deux façons… + Si on ne sait pas ça. Il nous reste à regarder des $5^m$ ou plutôt $5^{2m}$. - $(a^2 + b^2) (2^2 + 1^2) = (2a - b)^2 + (2b + a)^2$ $=(2b-a)^2 + (2a+b)^2$ - - !! + On a $5 = 4 + 1$ donc $5 = (2-i) (2+i)$, donc $5^{2m}$ a plein de façon de s'écrire comme un produit. Il faut montrer qu'elles donnent des décompositions distinctes… #+END_proof @@ -3739,7 +4038,7 @@ Soit $f\colon \R^{+*} \ra \R^{+*}$ de classe $\mc C^{\i}$ telle que $f(x) \ra 0$ 1. Montrer que la suite $(x_n)_{n\geq 1}$ est croissante. 1. Soit $n \in \N^*$. Montrer que $x^n f^{(n)}(x) \underset{x \ra 0^+}{\longrightarrow} 0$. 1. On pose $g(x) = \frac{f(x)}{x}$ pour tout $x$ \gt 0. Montrer que, pour tout $n \ge 0$, il existe $a_{n,0},\ldots,a_{n,n}\in\Z$ tels que $g^{(n)}(x)=\sum_{k=0}^n a_{n,k}\frac{f^{(n-k)}(x)}{x^{k+1}}$ pour tout x\gt 0. - 1. Montrer que, pour tout $n \ge 0$, $(-1)^n g^{(n)}(x) \gt 0$ pour tout $x$ \gt 0. + 1. Montrer que, pour tout $n \ge 0$, $(-1)^n g^{(n)}(x) \gt 0$ pour tout $x \gt 0$. #+end_exercice #+begin_exercice [X MP 2025 # 313] @@ -3765,9 +4064,20 @@ Soit $s: \R \ra \R$ telle que $(*): \forall x \in \R, s(x+1) = s(x) + \frac{1}{1 1. Que se passe-t-il si on remplace la condition $s(x) \xrightarrow[x \ra -\i]{} 0$ par la condition $s(x) \xrightarrow[x \ra +\i]{} 0$ ? #+end_exercice -#+begin_exercice [X MP 2025 # 316] - 1. Soit $f \in \mc C^0(\R, \R)$. Montrer que $f$ est affine si et seulement si, pour tout réel $x$, on a $\frac{f(x+h) + f(x-h) - 2f(x)}{h^2} \underset{h \ra 0^+}{\longrightarrow} 0$.b) Montrer que le résultat de la question précédente peut tomber en défaut sans hypothèse de continuité. +#+begin_exercice [X MP 2025 # 316] :todo: + 1. Soit $f \in \mc C^0(\R, \R)$. Montrer que $f$ est affine si et seulement si, pour tout réel $x$, on a $\frac{f(x+h) + f(x-h) - 2f(x)}{h^2} \underset{h \ra 0^+}{\longrightarrow} 0$. + 2. Et si $f$ n'est pas continue ? #+end_exercice +#+BEGIN_proof + 1. Si cette limite est vérifiée, on montre que $n(f(x + + \frac{\a}{n}) - f(x))\ra f(x+\a) - f(x)$ + + On suppose $f(0) = 0$. Alors $f(h) + f(-h) = o_0(h^2)$. + + + 2. Il suffit de prendre $f(x) = \a$ si $x\gt 0$ et $-\a$ sinon. +#+END_proof + #+begin_exercice [X MP 2025 # 317] Soit $F: \R \ra \R^{+*}$. On suppose qu'il existe $\alpha, \eta \gt 0$ tels que : @@ -3857,10 +4167,16 @@ Soit $f:[0,1]\ra\R$ continue. On pose $h:t\in[0,1]\mapsto \inf_{s\in[0,t]}f(s)$ #+begin_exercice [X MP 2025 # 323] Soit $\mc{P}$ l'ensemble des nombres premiers. -On pose $\Psi(x) = \sum_{\substack{p \in \mc{P}, \alpha \in \N^* \\ p^{\alpha} \leq x}} \ln p \text{ et } T(x) = \sum_{1 \leq n \leq x} \Psi\left(\frac{x}{n}\right)$. - 1. Montrer que $T(x) = \sum_{1 \le n \le n} \ln(n) = x \ln(x) x + O(\ln x)$ quand $x \ra +\i$. +On pose $\Psi(x) = \sum\limits_{\substack{p \in \mc{P}, \alpha \in \N^* \\ p^{\alpha} \leq x}} \ln p \text{ et } T(x) = \sum_{1 \leq n \leq x} \Psi\left(\frac{x}{n}\right)$. + 1. Montrer que $T(x) = \sum\limits_{1 \le n \le x} \ln(n) = x \ln(x) x + O(\ln x)$ quand $x \ra +\i$. 1. Montrer que $T(x) 2T\left(\frac{x}{2}\right) = \sum_{n=0}^{\i} (-1)^{n-1} \Psi\left(\frac{x}{n}\right) = x \ln 2 + O(\ln x)$. #+end_exercice +#+BEGIN_proof + 1. On a $\ln (n) = \sum_{p\in\mc P,\,\a ; p^{\a}\mid n} \ln p$, donc $\sum_{1\leq n\leq x}\ln n = \sum_{p, \a} \ln p \times \lfloor \frac{x}{p^{\a}}\rfloor$ + + et $\sum_{1\leq n\leq x \Psi(\frac{x}{n})} =$. +#+END_proof + # ID:nil # Facile #+begin_exercice [X MP 2025 # 324] @@ -3970,16 +4286,18 @@ $$g(x) = 1 - \frac{x}{2} + \sum_{n=1}^{+\i} a_n x^{2n}$$ 1. Pour $n \in \N^*$, donner une expression de $\zeta(2n)$ en fonction de $a_n$. Ind. On pourra considérer g(ix) pour $x \in \R$. #+end_exercice +# ID:8873 #+begin_exercice [X MP 2025 # 336] -Soit $f \in C^0([0,1], \R)$. - -Si $t \ge 0$, on pose $g_t \colon x \in [0, 1] \mapsto \inf \{ f(y) + t | y - x |, y \in [0, 1] \}$. +Soit $f \in C^0([0,1], \R)$. Si $t \ge 0$, on pose $g_t \colon x \in [0, 1] \mapsto \inf \{ f(y) + t | y - x |, y \in [0, 1] \}$. 1. Si $t \ge 0$, montrer que $g_t$ est une fonction continue. - 1. Soit $x \in [0, 1]$. Montrer que la suite $(g_n(x))_{n \ge 0}$ est croissante et qu'elle converge vers -f(x). - 1. Montrer que $(g_n)_{n\geq 0}$ converge uniformément vers $f$ sur [0,1]. + 1. Soit $x \in [0, 1]$. Montrer que la suite $(g_n(x))_{n \ge 0}$ est croissante et qu'elle converge vers $f(x)$. + 1. Montrer que $(g_n)_{n\geq 0}$ converge uniformément vers $f$ sur $[0,1]$. #+end_exercice +#+BEGIN_proof + 1. +#+END_proof + # ID:8764 #+begin_exercice [X MP 2025 # 337] @@ -4082,7 +4400,6 @@ En termes de distances : $\lN n (x,y) + m (0, 1)\rN^2 = nx^2 + (ny-m)^2 = n^2 ( Donc $(\sqrt{n}, 0)$ et $(0, 1)$. De là, ça ne doit plus être très dur. #+END_proof - ** Probabilités # ID:8450 @@ -5006,45 +5323,6 @@ Une randonneuse doit choisir un emplacement pour poser sa tente. Elle dispose de 1. Traiter le cas $N=3$. 2. (Question donnée après avoir fini Q1) On considère la stratégie suivante : la randonneuse parcourt les $k$ premiers emplacements sans s'arrêter, et à partir du $k+1$-ième, elle s'arrête dès qu'elle en trouve un meilleur que les précédents. Quel est le meilleur $k$ (asymptotiquement)? #+END_exercice -#+BEGIN_proof - 1. On a trois stratégies: - - s'arrêter systématiquement en position 1, - - s'arrêter en position 2 uniquement quand elle est meilleure que la première. - - s'arrêter systématiquement en position 3. - - Il y a 6 possibilité équiprobables pour l'ordre de la qualité des emplacements. Il nous suffit de dénombrer. - - On est dans la situation $(i, j, k)$ si l'emplacement $i$ est le meilleur, $j$ le deuxième meilleur et $k$ le moins bon. - - Les situations de succès de la première stratégie sont $(1,2,3)$ et $(1,3,2)$. La première stratégie est un succès avec probabilité $\frac{1}{3}$. - - Les situations de succès de la seconde stratégie sont $(2,1,3)$ et $(2,3,1)$ et $(3,1,2)$ (on ignore la position 2 car elle est moins bonne que la première). La deuxième stratégie est un succès avec probabilité $\frac{1}{2}$. - - Les situations de succès de la troisième stratégie sont $(3,1,2)$ et $(3,2,1)$. La troisième stratégie est un succès avec probabilité $\frac{1}{3}$. - 2. On note $s$ le classement des emplacements (le meilleur étant le mieux noté) qui nous donne une permutation de $\db{1, N }$ (on n'a pas d'ex-aequo). -Ainsi, si le meilleur emplacement est celui de numéro $i, s(i)=N$. L'emplacement $i$ est meilleur que l'emplacement $j \neq i$ si $s(j)\gt s(j)$. -Tous les classements sont supposés équiprobables et la probabilité que l'emplacement $i$ soit le meilleur vaut $\frac{1}{N}$ (il y a $(N-1)$ ! permutations $s$ de $\db{1, N }$ telles que $s(i)=N$). -On suppose que l'on igore les $k$ premiers emplacements ( $k \leq N-1$) et que l'on choisit le premier emplacement meilleur que les $k$ premiers (si cela existe, sinon, on choisit le dernier emplacement et on aura perdu). -Par formule des probabilités totales, la probabilité cherchée de l'événement $S_k$ de succès est - -$$\mathbb{P}\left(S_k\right)=\sum_{i=1}^N \mathbb{P}(s(i)=N) \mathbb{P}\left(S_k \mid s(i)=N\right)$$ - - -Il ne peut y avoir succès que si le meilleur emplacement est en position $i \geq k+1$ et donc - -$$\mathbb{P}\left(S_k\right)=\frac{1}{N} \sum_{i=k+1}^N \mathbb{P}\left(S_k \mid s(i)=N\right)$$ - - -On suppose que $s(i)=N$ et on aura un succès si le maximum parmi $s(1), \ldots, s(i-1)$ se trouve parmi les $k$ premières positions. Comme le maximum parmi les $i-1$ premiers se trouve demanière équiprobable à chaque position, la probabilité de succès sachant $s(i)=N$ est $\frac{k}{i-1}$. On a donc - -$$\mathbb{P}\left(S_k\right)=\frac{1}{N} \sum_{i=k+1}^N \frac{k}{i-1}=\frac{k}{N} \sum_{i=k}^{N-1} \frac{1}{i}$$ - -Ceci est cohérent avec la question 1 : si on choisit $N=3$ et $k=1$, on obtient une probabilité $1 / 2$ de succès. -On cherche désormais un $k \in \db{1, N }$ maximisant cette quantité. Mais comme on veut un résultat asymptotique, on cherche un résultat pour $N$ grand. Une comparaison série intégrale donne - -$$\frac{k}{N} \ln \left(\frac{N-1}{k-1}\right) \leq \mathbb{P}\left(S_k\right) \leq \frac{k}{N} \ln \left(\frac{N}{k}\right)$$ - - -Moralement, il s'agit de maximiser $\frac{\ln (x)}{x}$. Cette fonction est maximale en $x=e$. On peut donc estimer que la stratégie optimale utilise un entier $k$ proche de $\frac{N}{e}$. La probabilité de succès est alors proche de $\frac{1}{e}$ (dont une valeur approchée est 0.36, on a donc plus de 1 chance sur 3 d'obtenir le meilleur emplacement). -Il est probable que l'examinateur demanderait, à ce niveau, une formalisation plus poussée. -#+END_proof # ID:nil #+BEGIN_exercice [ENS 25, SR] @@ -5056,31 +5334,6 @@ $$e^M=\left(\begin{array}{cc} * & \Phi_{A, B}(C) \\* & * \end{array}\right)$$ 2. Montrer que $\Phi_{A, B}$ est linéaire. 3. On suppose $A, B$ diagonalisables. Montrer que $\Phi_{A, B}$ est diagonalisable. #+END_exercice -#+BEGIN_proof - - 1. Un calcul par blocs montre que $e^M=\left(\begin{array}{cc} e^A & \Phi_{A, B}(C) \\ 0 & e^B \end{array}\right)$ - - 2. Une récurrence montre que $\forall p \in \N, M^k=\left(\begin{array}{cc} A^p & \sum_{j=0}^{p-1} A^j C B^{p-1-j} \\ 0 & B^p \end{array}\right)$ - - et ainsi, $\Phi_{A, B}(C)=\sum_{p=0}^{\i}\left(\frac{1}{p!} \sum_{j=0}^{p-1} A^j C B^{p-1-j}\right)$ - - Cette expression permet alors de prouver aisément la linéarité de $\Phi_{A, B}$. - 3. On suppose $A$ et $B$ diagonalisables. $B^T$ est aussi diagonalisable. - -On peut trouver une base $\left(X_1, \ldots, X_n\right)$ de $\R^n$ formée de vecteurs propres pour $A$ et de même ue base $\left(Y_1, \ldots, Y_k\right)$ de $\C^k$ formée de vecteurs propres pour $B^T$. -On note $\lambda_i$ la valeur propre associée à $X_i$ et $\mu_i$ celle associée à $Y_i$. On a alors - -$$A^j\left(X_u Y_v^T\right) B^{p-1-j}=\lambda_u^j \mu_v^{p-1-j}\left(X_u Y_v^T\right)$$ - -et donc - -$$\Phi_{A, B}\left(X_u Y_v^T\right)=\sum_{p=0}^{\i}\left(\frac{1}{p!} \sum_{j=0}^{p-1} \lambda_u^j \mu_v^{p-1-j}\right)\left(X_u Y_v^T\right)$$ - - -La famille $\left(X_u Y_v^T\right)_{u \in[1, n], v \in[1, k]}$ est de cardinal $n k=\op{dim}\left(\M_{n, k}(\C)\right)$ et composée de vecteurs propres pour $\Phi_{A, B}$. -Pour conclure, il nous suffit de montrer que la famille est libre ou génératrice. Je montre que tout élément de la base canonique ( $E_{i, j}$) de $\M_{n, k}$ est combinaison linéaire des éléments de la famille $\left(X_u Y_v^T\right)_{u \in[1, n], v \in[1, k]}$. -Soient $i \in \db{1, n}$ et $j \in \db{1, k }$, on a $E_{i, j}=U_i^T V_j$ où $\left(U_1, \ldots, U_n\right)$ et $\left(V_1, \ldots, V_k\right)$ sont les bases canoniques de $\C^n$ et $\C^k$. On peut décomposer $U_i$ sur les $X_u$ et $V_j$ sur les $X_v$ et on peut conclure. -#+END_proof # ID:8238 #+BEGIN_exercice [ENS SR 25] @@ -5088,16 +5341,6 @@ Montrer le caractère $C^{\i}$ sur $\R^2$ de la fonction définie par $$\forall x \neq y, f(x, y)=\frac{e^x-e^y}{x-y} \text { et } \forall x, f(x, x)=e^x$$ #+END_exercice -#+BEGIN_proof -Posons $g\colon t \mapsto e^{t(x-y)+y} \cdot g$ est de classe $C^1$ sur $[0,1]$ et par théorème fondamental - -$$g(1)-g(0)=\int_0^1 g'(t)\dt=(x-y) \int_0^1 e^{t(x-y)+y}\dt$$ - - -On en déduit que pour $x \neq y$, $\displaystyle f(x, y)=\int_0^1 e^{t(x-y)+y}\dt$, expression qui reste valable si $x=y$. - -On peut alors utiliser le théorème sur les intégrales à paramètres pour justifier que $f$ admet des dérivées partielles à tout ordre en tout point de $\R^2$ et que ces dérivées partielles sont continues sur $\R^2$ (ce que je ne fais pas ici). Ceci montre que $f$ est de classe $C^{\i}$ sur $\R^2$. -#+END_proof #+BEGIN_exercice [X 2025] Soit $\alpha\gt 0$. On définit @@ -5118,51 +5361,11 @@ Soit $E$ un espace préhilbertien de dimension infinie. Soit $K$ une partie de $ Montrer que $K$ est d'intérieur vide. Question supplémentaire : et si on remplace l'hypothèse "préhilbertien" par "normé" ? #+END_exercice -#+BEGIN_proof - $E$ étant de dimension infinie, on peut construire une famille orthonormée infinie $\left(e_n\right)_{n \in \N}$. Supposons, par l'absurde, $K$ d'intérieur non vide. Il existe donc $a \in E$ et $r\gt 0$ tel que $\bar{B}(a, r) \subset K$. Pour $n \in \N$, l'ensemble $\left\{t \geq 0, a+t e_n \in K\right\}$ est non vide (il contient $r$) et majoré (car $K$ est borné). Il possède une borne supérieure $t_n \geq r$ et la suite $\left(t_n\right)$ est majorée (toujours car $K$ est borné). - -Pour tout $n$, on montre que $u_n=a+n e_n$ est dans la frontière de $K$. On peut en extraire une sous-suite $\left(u_{\phi(n)}\right)$ qui converge et on note $\ell \in E$ sa limite. Comme $\left(t_n\right)$ est bornée, quitte à extraire encore, on peut aussi supposer $t_{\phi(n)} \ra t \geq r$. -On a ainsi $e_{\phi(n)} \ra \frac{1}{t}(\ell-a)$. -En développant le carré scalaire, on a par ailleurs $\left\|e_{\phi(n)}-e_{\phi(n+1)}\right\|^2=2$ qui ne tend pas vers 0 et on a une contradiction. - -Le seul endroit où on utilise le caractère préhilbertien de $E$, c'est pour conclure à la fin (avant, on peut se contenter d'avoir une suite ( $e_n$) normée). -Pour généraliser le résultat, il suffit de montrer que l'on peut construire une suite ( $e_n$) de vecteurs normés tels que $\forall p \neq q,\left\|e_p-e_q\right\| \geq k$ où $k\gt 0$ est une constante (indépendante de $p$ et $q$). -On fait la construction par récurrence. On choisit pour $e_0$ un vecteur normé arbitraire (il en existe un). -Supposons $e_0, \ldots, e_n$ construits tels que $\left\|e_k\right\|=1$ et $\forall p \neq q,\left\|e_p-e_q\right\| \geq 1$. Notons $F$ l'espace engenré par $e_0, \ldots, e_n$. -$E$ étant de dimension finie, on peut trouver $a \in E \setminus F$. Il existe, par définition de la borne inférieure, une suite ( $b_n$) d'éléments de $F$ telle que $\left\|a-b_n\right\| \ra d(a, F)$. La suite ( $b_n$) est bien sûr bornée dans $F$ de dimension finie et on peut en extraire une sous-suite qui converge dans $F$. En notant $b \in F$ la limite de cette extraite, on a $\|a-b\|=d(a, F)\gt 0$ (puisque $F$ est fermé, cette distance est $\gt 0$). -Posons $c=\frac{a-b}{\|a-b\|}$; on a $\|c\|=1$ et - -$$d(c, F)=\inf_{x \in F}\|c-x\|=\inf_{x \in F}\left|\frac{a-b}{\|a-b\|}-x\right|=\frac{1}{\|a-b\|} \inf_{x \in F}\|a-b-\| a-b\|x\|$$ - - -Quand $x$ décrit $F, b+\|a-b\| x$ décrit aussi $F$ et donc - -$$d(c, F)=\frac{1}{\|a-b\|} \inf_{x \in F}\|a-x\|=\frac{1}{\|a-b\|} d(a, F)=1$$ - -On donc trouvé $c \in E$ tel que $\|c\|=1=d(c, F)$ ce qui permet la construction récurrente. -#+END_proof #+BEGIN_exercice [ENS 2025, MPI] Dans $\mc{S}_n^{++}(\R)$, on définit la relation d'ordre strict $\gt : A\gt B \Longleftrightarrow A-B \in \mc{S}_n^{++}(\R)$. Montrer que l'application $A \mapsto A^{-1}$ est décroissante sur $\mc{S}_n^{++}(\R)$. #+END_exercice -#+BEGIN_proof -On se donne $A, B \in \mc{S}_n^{++}(\R)$ telles que $A\lt B$, c'est-à-dire $\op{Sp}(B-A) \subset \R^{+*}$, et on veut montrer que $B^{-1}\lt A^{-1}$. - -Supposons que $B=I_n$. On a alors $\op{Sp}(A) \subset ] 0,1[$. Ainsi $\op{Sp}\left(A^{-1}\right) \subset ] 1,+\i [$ et donc $\op{Sp}\left(A^{-1}-I_n\right) \subset \R^{+*}$. Ceci montre que $B^{-1}\lt A^{-1}$. - -Dans le cas général, on montre aisément qu'il existe une (unique) matrice $C \in \mc{S}_n^{++}(\R)$ telle que $C^2=B. A\lt B$ donne - -$$\forall X \neq 0, X^T $A$ X\lt X^T B X=(C X)^T(C X)$$ - - -On se donne $Y \neq 0$ et on utilise ceci avec $X=C^{-1} Y$ pour obtenir - -$$\forall Y \neq 0, Y^T\left(C^{-1} $A$ C^{-1}\right) Y\lt Y^T Y$$ - - -Comme $C^{-1} $A$ C^{-1}$ est définie positive, on est ramenés au premier cas et $I_n\lt C^{-1} $A$ C^{-1}$ et comme plus haut on en déduit que $B^{-1}=\left(C^{-1}\right)^2\lt A^{-1}$. -#+END_proof #+BEGIN_exercice [X 2025] Soit $d \in \N^*$. On note $\quad\displaystyle f\colon z \mapsto \sum_{k \leq d, k \in \Z} c_k z^k$, et on suppose que $f$ est définie sur le complémentaire d'un disque centré en 0. @@ -5171,58 +5374,13 @@ On suppose également que $c_1, \ldots, c_d \in \Q$ et qu'il existe une infinit 2. Montrer que $\forall k\lt 0, c_k=0$. 3. Autres questions non abordées. #+END_exercice -#+BEGIN_proof - 1. Par hypothèse, il existe $r\gt 0$ tel que $f(z)$ existe quand $|z| \geq r$. On en déduit que $\sum\left(c_{-k} x^k\right)_{k \geq 0}$ est une série entière de rayon de convergence $\geq \frac{1}{r}\gt 0$ (la série converge si $0\lt |x|\lt 1 / r$). En particulier (la somme est continue sur $B(0,1 / r)$) - -$$\lim_{z \ra 0} \sum_{k=0}^{\i} c_{-k} z^k=0$$ - - -On peut trouver une suite ( $x_n$) d'entiers relatifs telle que $\left|x_n\right| \ra+\i$ et $\forall n, f\left(x_n\right) \in \Z$ c'est à dire - -$$c_d x_n^d+\cdots+c_1 x_n+\sum_{k=0}^{\i} \frac{c_{-k}}{x_n^k} \in \Z$$ - - -Notons $\mu$ le ppcm des dénominateurs de $c_1, \ldots, c_d$ (qui sont rationnels et s'écrivent $c_k=\frac{a_k}{b_k}$ avec $a_k \in \Z$ et $b_k \in \N^*$). On a alors - -$$\mu\left(c_d x_n^d+\cdots+c_1 x_n\right) \in \Z \text { et } \lim_{n \ra+\i} \mu \sum_{k=0}^{\i} \frac{c_{-k}}{x_n^k}=c_0 \mu$$ - -$\mu \sum_{k=0}^{\i} \frac{c-k}{x_n^k}$ est ainsi le terme général d'une suite d'entiers qui converge vers $c_0 \mu$ et comme $\Z$ est fermé, $c_0 \mu \in \Z$. Ainsi $c_0 \in \Q$. - 2. On reprend les notations précédentes et, quitte à extraire et à changer $c_k$ en $(-1)^k c_k$, on peut supposer disposer d'une suite d'entiers $\left(x_n\right)_{n \in \N}$ strictement croissante et telle que $\forall n f\left(x_n\right) \in \Z$. On continue d'utiliser la notation $\mu$. -Pour montrer la nullité des $c_k$ pour $k\lt 0$, il nous suffit de montrer que la somme $g(z)$ de la série entière $\sum\left(\mu c_{-k} x^k\right)_{k \geq 0}$ est constante (et d'utiliser l'unicité d'un DSE). -On sait que - -$$\forall n \in \N, \sum_{k=0}^{\i} \frac{\mu c_{-k}}{x_n^k} \in \Z$$ - -et qu'on a le terme général d'une suite convergente de limite $\mu c_0$. Cette suite est donc stationnaire à partir d'un certain rang. On en déduit que $g(z)-\mu c_0$ s'annule une infinité de fois. Par le principe des zéros isolés, cette fonction est nulle et les $c_{-k}$ pour $k\lt 0$ sont nuls. -#+END_proof +# ID:nil #+BEGIN_exercice [X 2025] Quels sont les entiers $n \in \N^*$ tels que $\cos \left(\frac{2 \pi}{n}\right) \in \Q$ ? #+END_exercice #+BEGIN_proof -On montre tout d'abord avec une récurrence que pour tout $k \in \N$, il existe un polynôme unitaire $P_k$ de $\Z[X]$ tel que - -$$\forall x \in \R, \quad P_k(2 \cos (x))=2 \cos (k x)$$ - -- C'est vrai aux rangs 0 et 1 avec $P_0=1$ et $P_1=X$. -- On suppose le résultat vrai jusqu'à un rang $k \geq 1$ et on écrit que - -$$\cos ((k+1) x)+\cos ((k-1) x)=2 \cos (k x) \cos (x)=P_k(2 \cos (x)) \cos (x)$$ - -et il suffit de poser $P_{k+1}=X P_k-P_{k-1}$. - -Soit $n$ un tel entier. $\cos \left(\frac{2 \pi}{n}\right)$ est donc un rationnel et s'écrit $\frac{a}{b}$ avec $a \wedge b=1$. -On a $P_n\left(\frac{2 a}{b}\right)=2$ et comme $P_n$ est à coefficients entiers et unitaire, on trouve des entiers $c_k$ tels que - -$$\frac{2^n a^n}{b^n}+\sum_{k=0}^{n-1} c_k \frac{2^k a_k}{b^k}=0$$ - - -On en déduit que - -$$2^n a_n+c_{n-1} 2^{n-1} a_{n-1} b+\cdots+c_1 2 a b^{n-1}+a_0 b^n=0$$ - -$b \mid 2^n a_n$ et $b \wedge a=1$ donc $b \mid 2^n$. Il existe $k \in \db{0, n }$ tel que $b=2^k$. Si $k \geq 2, b$ est pair et tous les termes ci-dessus sauf le premier sont divisibles par $2^{n+1}$ ce qui entraîne que $a$ est multiple de 2 et contredit $a \wedge b=1$. -Ainsi $b \in\{1,2\}$ et $2 \cos \left(\frac{2 \pi}{n}\right) \in \Z$ puis $\cos \left(\frac{2 \pi}{n}\right) \in\{-1,-1 / 2,0,1 / 2,1\}$ et (facile) $n \in\{1,2,3,4,5,6\}$. On vérifie que $1,2,3,4$ et 6 conviennent (valeurs de cosinus $1,-1,-1 / 2,0,1 / 2$). Il reste à tester si $\cos \left(\frac{2 \pi}{5}\right)$ est rationnel. Le lecteur montrera que $\cos \left(\frac{2 \pi}{5}\right)=\frac{-1+\sqrt{5}}{2}$ et que $n=5$ ne convient pas. +Via un polynôme $P_k$ tel que $\forall x \in \R, \quad P_k(2 \cos (x))=2 \cos (k x)$. #+END_proof @@ -6603,7 +6761,7 @@ b) Donner un équivalent de $u_n$, cardinal de $\{(x,y,z) \in \db{1,n}^3, xy=z\} #+begin_exercice [Mines MP 2025 # 644] Soient $\lambda \in \mathbb{R}$ et $(u_n)_{n \geq 0}$ une suite d'éléments de $\mathbb{R}^{+*}$. -On suppose que $\frac{u_{n+1}}{u_n} = 1 \frac{\lambda}{n} + O\left(\frac{1}{n^2}\right)$. Déterminer la nature de la série $\sum u_n$. +On suppose que $\frac{u_{n+1}}{u_n} = 1-\frac{\lambda}{n} + O\left(\frac{1}{n^2}\right)$. Déterminer la nature de la série $\sum u_n$. #+end_exercice #+begin_exercice [Mines MP 2025 # 645] @@ -6613,9 +6771,8 @@ Soit $\alpha \in \mathbb{R}$. Nature de $\sum_{n=0}^{\infty} (-1)^n \frac{\sin(\ #+begin_exercice [Mines MP 2025 # 646] Soit $\alpha$ un réel > 1. Pour tout $n \geq1$, on pose $u_n = \sum_{n=0}^{\infty} \frac{1}{k^{\alpha}}$. -a) Vérifier la bonne définition de $u_n$, et en donner un équivalent quand $n \to +\infty$. b) On pose $v_n = \frac{u_{n^2}}{u_n}$ pour tout $n \geq 1$. Étudier la convergence des séries $\sum v_n$ et - - $\sum (-1)^n v_n$. + a) Vérifier la bonne définition de $u_n$, et en donner un équivalent quand $n \to +\infty$. + b) On pose $v_n = \frac{u_{n^2}}{u_n}$ pour tout $n \geq 1$. Étudier la convergence des séries $\sum v_n$ et $\sum (-1)^n v_n$. #+end_exercice #+begin_exercice [Mines MP 2025 # 647] @@ -6626,15 +6783,37 @@ Pour $n \in \N^*$, on pose $u_n = \frac{\sqrt{n!}}{\left(1+\sqrt{1}\right)\left( On admet que $\sum_{n=0}^{+\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$. Démontrer la convergence et calculer la somme : $\sum_{n=1}^{+\infty} \frac{(-1)^n}{2n-1} \sum_{k=-1}^{+\infty} \frac{(-1)^{k-1}}{2k-1}$. #+end_exercice +# ID:8800 #+begin_exercice [Mines MP 2025 # 649] -a) Montrer que : $\sum_{k=0}^{n} (-1)^{\lfloor \sqrt{k} \rfloor} = O(\sqrt{n})$. - -b) Soit $z \in \mathbb{U}$. Montrer que $\sum \frac{(-1)^{\lfloor \sqrt{k} \rfloor}}{k} z^k$ est une série convergente. + 1. Montrer que : $\sum_{k=0}^{n} (-1)^{\lfloor \sqrt{k} \rfloor} = O(\sqrt{n})$. + 2. Soit $z \in \mathbb{U}$. Montrer que $\sum \frac{(-1)^{\lfloor \sqrt{k} \rfloor}}{k} z^k$ est une série convergente. #+end_exercice +# ID:8801 #+begin_exercice [Mines MP 2025 # 650] Soit $(u_n)_{n\geq 0}\in (\mathbb{R}^+)^{\N}$. On pose, pour $n\in \N$, $v_n=\frac{1}{1+n^2u_n}$. Montrer que si la série $\sum v_n$ converge alors la série $\sum u_n$ diverge. #+end_exercice +#+begin_proof +Par l'absurde, supposons que la série $\sum u_n$ converge. +D'après l'inégalité de Cauchy-Schwarz, on a pour tout $n\ge 1$ : +$$n = \sum_{k=1}^n 1 = \sum_{k=1}^n \sqrt{v_k} \sqrt{1+k^2u_k} \le \left(\sum_{k=1}^n v_k \right)^{1/2} \left(\sum_{k=1}^n (1+k^2u_k) \right)^{1/2}.$$ + +La série $\sum v_n$ étant convergente à termes positifs, notons $V = \sum_{k=1}^{+\infty} v_k > 0$. En élevant au carré, on obtient : +$$n^2 \le V \left(n + \sum_{k=1}^n k^2 u_k\right) \implies \frac{1}{n^2}\sum_{k=1}^n k^2 u_k \ge \frac{1}{V} - \frac{1}{n}.$$ + +Or, puisque $\sum u_n$ converge, on peut estimer cette somme par une transformation d'Abel (ou le lemme de Kronecker). Posons $R_k = \sum_{j=k}^{+\infty} u_j$. On a $R_k \xrightarrow[k\to \infty]{} 0$ et $u_k = R_k - R_{k+1}$. +Alors : +$$\sum_{k=1}^n k^2 u_k = \sum_{k=1}^n k^2 (R_k - R_{k+1}) = R_1 - n^2 R_{n+1} + \sum_{k=2}^n (2k-1) R_k.$$ + +Comme $u_n \ge 0$, on a $R_{n+1} \ge 0$. On en déduit que : +$$0 \le \frac{1}{n^2}\sum_{k=1}^n k^2 u_k \le \frac{R_1}{n^2} + \frac{n^2-1}{n^2} \frac{\sum_{k=2}^n (2k-1) R_k}{\sum_{k=2}^n (2k-1)}.$$ + +D'après le lemme de Cesàro (les poids $2k-1$ étant strictement positifs et de somme divergente), la moyenne pondérée de la suite $(R_k)$ tend vers $0$ car $R_k \to 0$. +Ainsi, $\frac{1}{n^2}\sum_{k=1}^n k^2 u_k \xrightarrow[n\to \infty]{} 0$. + +En passant à la limite dans l'inégalité issue de Cauchy-Schwarz, on trouve $0 \ge \frac{1}{V} > 0$, ce qui est absurde. La série $\sum u_n$ diverge donc. +#+end_proof + #+begin_exercice [Mines MP 2025 # 651] a) Montrer que la série $\sum \frac{2^{k+1}(k!)^2}{(2k+1)!}$ converge. On pose $\sigma = \sum_{k=1}^{+\infty} \frac{2^{k+1}(k!)^2}{(2k+1)!}$. @@ -6655,16 +6834,31 @@ d) Montrer que $\arctan(x) = \sum_{n=0}^{+\infty} \frac{(-1)^n}{2n+1} x^{2n+1}$ #+begin_exercice [Mines MP 2025 # 652] On pose $a_n = \frac{n! e^n}{n^n \cdot \sqrt{n}}$. -a) Montrer que $\sum (\ln(a_{n+1}) \ln(a_n))$ converge. +a) Montrer que $\sum (\ln(a_{n+1})- \ln(a_n))$ converge. b) Donner un équivalent de $\sum_{k=0}^{+\infty} \frac{1}{k^2}$ quand $n$ tend vers $+\infty$. c) Démontrer que $n! \sim \sqrt{2\pi n} \, n^n e^{-n} \left( 1 + \frac{1}{12n} + o\left(\frac{1}{n}\right) \right)$. #+end_exercice +# ID:8803 #+begin_exercice [Mines MP 2025 # 653] -a) Soit a > 0. Discuter de la nature de $\sum \frac{1}{(n!)^{a/n}}$ en fonction de a. -b) Soit $(v_n)_{n\geq 0}\in\mathbb{R}^{\N}$ telle que $\sum (n!)^{2/n}v_n^2$ converge. Montrer que $\sum v_n$ converge. + 1. Soit $a > 0$. Discuter de la nature de $\sum \frac{1}{(n!)^{a/n}}$ en fonction de a. + 2. Soit $(v_n)_{n\geq 0}\in\mathbb{R}^{\N}$ telle que $\sum (n!)^{2/n}v_n^2$ converge. Montrer que $\sum v_n$ converge. #+end_exercice +#+begin_proof + 1. D'après la formule de Stirling, $\ln(n!) = n\ln n - n + O(\ln n)$, donc + $\frac{a}{n}\ln(n!) = a\ln n - a + o(1)$ + On en déduit que $(n!)^{a/n} \sim n^a e^{-a}$, puis que : + $\frac{1}{(n!)^{a/n}} \sim \frac{e^a}{n^a}$ + Par le critère d'équivalence des séries à termes positifs, la série $\sum \frac{1}{(n!)^{a/n}}$ converge si et seulement si $a > 1$ (série de Riemann). + 2. On remarque que $|v_n| = \left((n!)^{1/n}|v_n|\right) \times \frac{1}{(n!)^{1/n}}$. Par l'inégalité de Cauchy-Schwarz sur les sommes partielles : + $$\sum_{k=1}^n |v_k| \leq \left(\sum_{k=1}^n (k!)^{2/k}v_k^2\right)^{1/2} \left(\sum_{k=1}^n \frac{1}{(k!)^{2/k}}\right)^{1/2}$$ + La première somme converge par hypothèse. La seconde converge également d'après la question précédente avec $a=2 > 1$. + La série $\sum |v_n|$ est donc à termes positifs et ses sommes partielles sont majorées : elle converge. L'absolue convergence impliquant la convergence, la série $\sum v_n$ converge. +#+end_proof + + +# ID:8802 #+begin_exercice [Mines MP 2025 # 654] Montrer que les séries $\sum_{n=1}^{\infty} \frac{\sin(\ln n)}{n}$ et $\sum_{n=1}^{\infty} \int_{n}^{n+1} \frac{\sin(\ln t)}{t} dt$ sont de même nature. @@ -7494,11 +7688,38 @@ $M(0) \in \mathcal{A}$. Montrer que : $\forall t \in \mathbb{R}^+, M(t) \in \mat #+begin_exercice [Mines MP 2025 # 773] Soit $A:[0,1]\to\mathcal{M}_n(\mathbb{R})$ une fonction de classe $\mathcal{C}^1$ telle que $\forall t \in [0, 1], A^2(t) - 5A(t) + 6I_n = 0$. -a) Montrer qu'il existe une fonction $P:[0,1]\to \mathrm{GL}_n(\mathbb{R})$ telle que, pour tout $t\in[0,1]$, -$A(t) = P(t)A(0)P(t)^{-1}$. b) Soient $t \in [0, 1], \lambda \in \mathbb{R}$ et $X \in \mathcal{M}_{n,1}(\mathbb{R})$ tels que $A(t)X = \lambda X$. -Montrer que $A(t)A'(t)X = (5 \lambda)A'(t)X$. -c) Montrer qu'il existe une fonction $P:[0,1]\to \mathrm{GL}_n(\mathbb{R})$ de classe $\mathcal{C}^1$ telle que $\forall t\in[0,1]$, $A(t) = P(t)A(0)P(t)^{-1}$. + 1. Montrer qu'il existe une fonction $P:[0,1]\to \mathrm{GL}_n(\mathbb{R})$ telle que, pour tout $t\in[0,1]$, $A(t) = P(t)A(0)P(t)^{-1}$. + 2. Soient $t \in [0, 1], \lambda \in \mathbb{R}$ et $X \in \mathcal{M}_{n,1}(\mathbb{R})$ tels que $A(t)X = \lambda X$. Montrer que $A(t)A'(t)X = (5 -\lambda)A'(t)X$. + 3. Montrer qu'il existe une fonction $P:[0,1]\to \mathrm{GL}_n(\mathbb{R})$ de classe $\mathcal{C}^1$ telle que $\forall t\in[0,1]$, $A(t) = P(t)A(0)P(t)^{-1}$. #+end_exercice +#+begin_proof + 1. Le polynôme $X^2 - 5X + 6 = (X-2)(X-3)$ est annulateur de $A(t)$. La matrice $A(t)$ est donc diagonalisable et son spectre est inclus dans $\{2, 3\}$. + Notons $m_2(t)$ et $m_3(t)$ les multiplicités respectives des valeurs propres $2$ et $3$. On a alors : + $$\mathrm{Tr}(A(t)) = 2m_2(t) + 3m_3(t) = 2(n - m_3(t)) + 3m_3(t) = 2n + m_3(t)$$ + L'application $t \mapsto A(t)$ étant continue, $t \mapsto \mathrm{Tr}(A(t))$ l'est aussi. Comme elle est à valeurs entières, elle est constante sur l'intervalle $[0,1]$. + Il en découle que $m_3(t)$ et $m_2(t)$ sont constantes. Deux matrices diagonalisables ayant le même spectre et les mêmes multiplicités étant semblables, $A(t)$ est semblable à $A(0)$ pour tout $t\in[0,1]$. + + 2. En dérivant la relation $A(t)^2 - 5A(t) + 6I_n = 0$, on obtient : + $A'(t)A(t) + A(t)A'(t) - 5A'(t) = 0$ + Évaluons cette expression sur le vecteur $X$ : + $A'(t)A(t)X + A(t)A'(t)X - 5A'(t)X = 0$ + Comme $A(t)X = \lambda X$, on a $A'(t)A(t)X = \lambda A'(t)X$, ce qui donne : + $$\lambda A'(t)X + A(t)A'(t)X - 5A'(t)X = 0 \implies A(t)A'(t)X = (5 - \lambda)A'(t)X$$ + + 3. Posons $M(t) = A'(t)A(t) - A(t)A'(t)$. L'application $A$ étant de classe $\mathcal{C}^1$, $M$ est continue. + Soit $X$ un vecteur propre de $A(t)$ associé à la valeur propre $\lambda \in \{2,3\}$. D'après la question 2, $A'(t)X$ appartient à l'espace propre associé à $5-\lambda$. Calculons : + $$M(t)X = A'(t)(\lambda X) - A(t)A'(t)X = \lambda A'(t)X - (5-\lambda)A'(t)X = (2\lambda - 5)A'(t)X$$ + Évaluons maintenant l'action du commutateur $[M(t), A(t)]$ sur $X$ : + $$\begin{aligned}(M(t)A(t) - A(t)M(t))X &= M(t)(\lambda X) - A(t)\left((2\lambda - 5)A'(t)X\right) \\ &= \lambda(2\lambda - 5)A'(t)X - (2\lambda - 5)(5-\lambda)A'(t)X \\ &= (2\lambda - 5)(\lambda - 5 + \lambda)A'(t)X \\ &= (2\lambda - 5)^2 A'(t)X\end{aligned}$$ + Pour $\lambda \in \{2,3\}$, on a toujours $(2\lambda - 5)^2 = (\pm 1)^2 = 1$. Ainsi, sur tout vecteur propre $X$, $(M(t)A(t) - A(t)M(t))X = A'(t)X$. + Comme $A(t)$ est diagonalisable, cette égalité est vraie sur tout $\R^n$ : + $M(t)A(t) - A(t)M(t) = A'(t)$ + Soit alors $P$ l'unique solution du problème de Cauchy $P'(t) = M(t)P(t)$ avec $P(0) = I_n$. + + L'application $M$ étant continue, $P$ est de classe $\mathcal{C}^1$. D'après le théorème de Cauchy-Lipschitz linéaire, on a $P(t) \in \mathrm{GL}_n(\R)$ pour tout $t$. + + Considérons $B(t) = P(t)^{-1}A(t)P(t)$. En dérivant cette relation (et sachant que $(P^{-1})' = -P^{-1}P'P^{-1} = -P^{-1}M$), on obtient : + $$B'(t) = P(t)^{-1}\left(-M(t)A(t) + A'(t) + A(t)M(t)\right)P(t) = 0$$ + La fonction $B$ est donc constante, égale à $B(0) = A(0)$. On en déduit bien $A(t) = P(t)A(0)P(t)^{-1}$ pour tout $t\in[0,1]$. +#+end_proof #+begin_exercice [Mines MP 2025 # 774] On considère la fonction $f: \mathbb{R}^2 \to \mathbb{R}, (x,y) \mapsto \begin{cases} (x^2 + y^2)^x \text{ si } (x,y) \neq (0,0) \\ 1 \text{ si } x = y = 0. \end{cases}$. @@ -10870,20 +11091,21 @@ b) Montrer que $f(x) \xrightarrow[x \to +\infty]{} +\infty$ et $f'(x) \xrightarr #+end_exercice #+begin_exercice [Centrale MP 2025 # 1214] -Soit $(a_n) \in \mathbb{R}^{\N}$ telle que la série $\sum a_n$ converge. Soit $S: x \mapsto \sum_{n=1}^{\infty} a_n \cos\left(\frac{x}{n}\right)$. +Soit $(a_n) \in \mathbb{R}^{\N}$ telle que la série $\sum a_n$ converge. Soit $S\colon x \mapsto \sum_{n=1}^{\infty} a_n \cos\left(\frac{x}{n}\right)$. On suppose que $S$ a une limite réelle $\ell$ en $+\infty$. On souhaite montrer que la suite $(a_n)$ est nulle. -a) i) Énoncer l'inégalité de Taylor-Lagrangeqà un ordre quelconque. -ii) Montrer que $S$ est bien définie sur $\mathbb{R}$. -b) On suppose dans cette question que la série $\sum a_n$ converge absolument et que $\ell=0$. - - i) Montrer que $S$ est continue. - - ii) Soit $m \in \N^*$. On pose $I: T \in \mathbb{R}^{+*} \mapsto \frac{1}{T} \int_0^T S(x) \cos\left(\frac{x}{m}\right) dx$. + 1. + 1. Énoncer l'inégalité de Taylor-Lagrangeqà un ordre quelconque. + 2. Montrer que $S$ est bien définie sur $\mathbb{R}$. + 2. On suppose dans cette question que la série $\sum a_n$ converge absolument et que $\ell=0$. + - Montrer que $S$ est continue. + - Soit $m \in \N^*$. On pose $I\colon T \in \mathbb{R}^{+*} \mapsto \frac{1}{T} \int_0^T S(x) \cos\left(\frac{x}{m}\right) dx$. -Montrer que $\lim_{T \to +\infty} I(T) = 0$. + Montrer que $\lim_{T \to +\infty} I(T) = 0$. -iii) Montrer que $a_m = 0$. -c) Traiter le cas général. + - Montrer que $a_m = 0$. + 3. Traiter le cas général. #+end_exercice #+begin_exercice [Centrale MP 2025 # 1215] @@ -12384,9 +12606,11 @@ Soient $a, b, c \in \mathbb{R}$. Pour $n \in \N^*$, on pose $u_n = a \ln(n) + b On pose, pour $n \geq 3$, $u_n = \ln\left(\frac{n^4 2n^3 + 2n 1}{n^4 2n^3}\right)$. Étudier la nature de la série de terme général $u_n$ et calculer sa somme en cas de convergence. #+end_exercice +# ID:8804 #+begin_exercice [IMT # 1396] Soit $\sigma: \N \to \N$ telle que $\sigma(3n) = 4n$, $\sigma(3n+1) = 4n+2$ et $\sigma(3n+2) = 2n+1$ pour tout $n \in \N$. -a) Montrer que $\sigma$ est bijective. b) On pose $u_n = \frac{(-1)^n}{n}$ et $v_n = u_{\sigma(n)}$ pour tout $n \geq1$. Montrer que les séries $\sum u_n$ et $\sum v_n$ sont convergentes, et calculer leurs sommes. + 1. Montrer que $\sigma$ est bijective. + 2. On pose $u_n = \frac{(-1)^n}{n}$ et $v_n = u_{\sigma(n)}$ pour tout $n \geq1$. Montrer que les séries $\sum u_n$ et $\sum v_n$ sont convergentes, et calculer leurs sommes. #+end_exercice #+begin_exercice [IMT # 1397] diff --git a/Exercices XENS MP 2025.pdf b/Exercices XENS MP 2025.pdf index 7c477fdac14fe615e6b536b926f459de260efeb7..0dc314d5af429b181cc9c653f39e0d959be3661d 100644 GIT binary patch delta 519589 zcmV(;K-<5swK0*)C=ex3L`E$!E;kA#(A34)3O1Kvw+j;lG&MIgml4YWD1Yr-?UUQK z(f{thLTx^TH?>f_19ZpFGkLK)>9|fByEC1f-Ki|e4ikwwi&AR$ukV5YC4yQ|r0()O zO?qEW(h&r(i+%a+VgWjNH=&b1KA>klKY#h)_fK;e+yNv^qeBTzR6u{?DB)hnvl z)s$&wvnS7A_CJGG&b9 zO8H4KKPfFt_Sx#4qwq~lVl*raq&dyV)l|{@59~wc`G4rlIHfu%AWlTv(!g8NP|j8# zs}ng;_j2$HmIHH64z!m8ag>8Uf=z(0Cflstx81flkqNHl{qp~!gaKSEcnvYoD#)WW zeB+$M=FRL)u_|^>9)GH?Fi4f(FaNJW3OZP0dO#1FXgPQ*aX^{7z=3WX8DWS_iO+Yo)%IwRK2>Rmt zZBgwuH{UODbVGG+hM*h9na#L9ZPm(5xxA4kit%>ds3Vb#0)M$!7OBf6J@X%R zs;~a`NqG5ncYGQ3B6Xg<*c3ZIh7lb=sXjz3R@5?afYG3Ta*P{Gk z>3r~fSNv4VQbgQM?VgKrS0%}tVKWtyNZ>FdG7poWNPh>=NavGQqd&OBtjQj^5A8op 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