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Question Number 212984 by Faetmaaa last updated on 27/Oct/24

$$\underset{―}{\mathbf{Notation}:}\mathrm{Soit}A\mathrm{une}\mathrm{partie}\mathrm{de}\mathbb{R}.\mathrm{On}\mathrm{appelle}indicatricedeA,$$$$\mathrm{not}\acute{\mathrm{e}e}{\chi }_A,{\mathrm{l}}^{\prime }\mathrm{application}x\{\begin{array}{l}1\mathrm{si}x\in A\\ 0\mathrm{sinon}\end{array}.$$$$$$$$1.\mathrm{Pour}k\mathrm{dans}{\mathbb{N}}^{\ast }\mathrm{notons}{f}_k:x{\left(\cos x\right)}^{2k}.$$$$\mathrm{Montrer}\mathrm{que}{\left(f_k\right)}_{k\in {\mathbb{N}}^{\ast }}\mathrm{converge}\mathrm{vers}{\chi }_{\pi \mathbb{Z}}.$$$$$$$$2.\mathrm{Soit}n\mathrm{un}\mathrm{param}\stackrel{`}{\mathrm{e}tre}\mathrm{fix}\acute{\mathrm{e}}\mathrm{dans}\mathbb{N}.$$$$\mathrm{Pour}k\mathrm{dans}{\mathbb{N}}^{\ast }\mathrm{notons}{g}_k:x{f}_k(n!\pi x).$$$$\mathrm{Montrer}\mathrm{que}{\left(g_k\right)}_{k\in {\mathbb{N}}^{\ast }}\mathrm{converge}\mathrm{vers}\mathrm{une}\mathrm{application}{g}^{\left(n\right)}\stackrel{`}{\mathrm{a}}$$$$\mathrm{d}\acute{\mathrm{e}terminer},\mathrm{d}\acute{\mathrm{e}pendante}\mathrm{du}\mathrm{param}\stackrel{`}{\mathrm{e}tre}n.\acute{\mathrm{E}crire}{g}^{\left(n\right)}\mathrm{sous}$$$$\mathrm{la}\mathrm{forme}{\chi }_A\mathrm{o}\stackrel{`}{\mathrm{u}}A\mathrm{est}\mathrm{une}\mathrm{partie}\mathrm{de}\mathbb{R}\stackrel{`}{\mathrm{a}}\mathrm{d}\acute{\mathrm{e}terminer}.$$$$$$$$3.\mathrm{Montrer}\mathrm{que}\mathrm{la}\mathrm{suite}\mathrm{de}\mathrm{fonctions}{\left(g^{\left(n\right)}\right)}_{n\in \mathbb{N}}\mathrm{converge}\mathrm{vers}{\chi }_{\mathbb{Q}}.$$$$$$$$4.\mathrm{Soit}x\mathrm{dans}\mathbb{R}.\mathrm{Calculer}\underset{n\to \mathrm{\infty }}{\lim }\left(\underset{k\to \mathrm{\infty }}{\lim }{\left(\cos (n!\pi x)\right)}^{2k}\right).$$

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