Oraux_CentraleSupelec_PSI__Juin_2018.ipynb

{
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   "cell_type": "markdown",
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    "toc": "true"
   },
   "source": [
    "# Table of Contents\n",
    " <p><div class=\"lev1 toc-item\"><a href=\"#Oraux-CentraleSupélec-PSI---Juin-2018\" data-toc-modified-id=\"Oraux-CentraleSupélec-PSI---Juin-2018-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Oraux CentraleSupélec PSI - Juin 2018</a></div><div class=\"lev2 toc-item\"><a href=\"#Remarques-préliminaires\" data-toc-modified-id=\"Remarques-préliminaires-11\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>Remarques préliminaires</a></div><div class=\"lev2 toc-item\"><a href=\"#Planche-160\" data-toc-modified-id=\"Planche-160-12\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>Planche 160</a></div><div class=\"lev2 toc-item\"><a href=\"#Planche-162\" data-toc-modified-id=\"Planche-162-13\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>Planche 162</a></div><div class=\"lev2 toc-item\"><a href=\"#Planche-166\" data-toc-modified-id=\"Planche-166-14\"><span class=\"toc-item-num\">1.4&nbsp;&nbsp;</span>Planche 166</a></div><div class=\"lev2 toc-item\"><a href=\"#Planche-168\" data-toc-modified-id=\"Planche-168-15\"><span class=\"toc-item-num\">1.5&nbsp;&nbsp;</span>Planche 168</a></div><div class=\"lev2 toc-item\"><a href=\"#Planche-170\" data-toc-modified-id=\"Planche-170-16\"><span class=\"toc-item-num\">1.6&nbsp;&nbsp;</span>Planche 170</a></div><div class=\"lev2 toc-item\"><a href=\"#Planche-172\" data-toc-modified-id=\"Planche-172-17\"><span class=\"toc-item-num\">1.7&nbsp;&nbsp;</span>Planche 172</a></div><div class=\"lev2 toc-item\"><a href=\"#Planche-177\" data-toc-modified-id=\"Planche-177-18\"><span class=\"toc-item-num\">1.8&nbsp;&nbsp;</span>Planche 177</a></div><div class=\"lev1 toc-item\"><a href=\"#À-voir-aussi\" data-toc-modified-id=\"À-voir-aussi-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>À voir aussi</a></div><div class=\"lev2 toc-item\"><a href=\"#Les-oraux---(exercices-de-maths-avec-Python)\" data-toc-modified-id=\"Les-oraux---(exercices-de-maths-avec-Python)-21\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span><a href=\"http://perso.crans.org/besson/infoMP/oraux/solutions/\" target=\"_blank\">Les oraux</a>   <em>(exercices de maths avec Python)</em></a></div><div class=\"lev2 toc-item\"><a href=\"#Fiches-de-révisions-pour-les-oraux\" data-toc-modified-id=\"Fiches-de-révisions-pour-les-oraux-22\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>Fiches de révisions <em>pour les oraux</em></a></div><div class=\"lev2 toc-item\"><a href=\"#Quelques-exemples-de-sujets-d'oraux-corrigés\" data-toc-modified-id=\"Quelques-exemples-de-sujets-d'oraux-corrigés-23\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>Quelques exemples de sujets <em>d'oraux</em> corrigés</a></div><div class=\"lev2 toc-item\"><a href=\"#D'autres-notebooks-?\" data-toc-modified-id=\"D'autres-notebooks-?-24\"><span class=\"toc-item-num\">2.4&nbsp;&nbsp;</span>D'autres notebooks ?</a></div>"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Oraux CentraleSupélec PSI - Juin 2018\n",
    "\n",
    "- Ce [notebook Jupyter](https://www.jupyter.org) est une proposition de correction, en [Python 3](https://www.python.org/), d'exercices d'annales de l'épreuve \"maths-info\" du [concours CentraleSupélec](http://www.concours-centrale-supelec.fr/), filière PSI.\n",
    "- Les exercices viennent de l'[Officiel de la Taupe](http://odlt.fr/), [2017](http://www.odlt.fr/Oraux_2018.pdf) (planches 162 à 177, page 23).\n",
    "- Ce document a été écrit par [Lilian Besson](http://perso.crans.org/besson/), et est disponible en ligne [sur mon site](https://perso.crans.org/besson/publis/notebooks/Oraux_CentraleSupelec_PSI__Juin_2018.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Remarques préliminaires\n",
    "- Les exercices sans Python ne sont pas traités.\n",
    "- Les exercices avec Python utilisent Python 3, [numpy](http://numpy.org), [matplotlib](http://matplotlib.org), [scipy](http://scipy.org) et [sympy](http://sympy.org), et essaient d'être résolus le plus simplement et le plus rapidement possible. L'efficacité (algorithmique, en terme de mémoire et de temps de calcul), n'est *pas* une priorité. La concision et simplicité de la solution proposée est prioritaire."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import integrate\n",
    "import numpy.random as rd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Planche 160"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- $I_n := \\int_0^1 \\frac{1}{(1+t)^n \\sqrt{1-t}} \\mathrm{d}t$ et $I_n := \\int_0^1 \\frac{1/2}{(1+t)^n \\sqrt{1-t}} \\mathrm{d}t$ sont définies pour tout $n$ car leur intégrande est continue et bien définie sur $]0,1[$ et intégrable en $1$ parce qu'on sait (par intégrale de Riemann) que $\\frac{1}{\\sqrt{u}}$ est intégrable en $0^+$ (et changement de variable $u = 1-t$).\n",
    "- On les calcule très simplement :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def I(n):\n",
    "    def f(t):\n",
    "        return 1 / ((1+t)**n * np.sqrt(1-t))\n",
    "    i, err = integrate.quad(f, 0, 1)\n",
    "    return i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def J(n):\n",
    "    def f(t):\n",
    "        return 1 / ((1+t)**n * np.sqrt(1-t))\n",
    "    i, err = integrate.quad(f, 0, 0.5)\n",
    "    return i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5eb317f0>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Valeurs de $I_n$')"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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7acC9fcuHe21LeTXwp4utSLI3yVySuSNHjgxe5UY2f/37wYNQ9cj17wa8pBEa6gnVJK8AdgO/tdj6qpqpqt1VtXtycnKYLz06Xv8uaQMa5BOq9wGn9y3v6LUdI8mLgSuA51fVt4dT3ibg9e+SNqBBRu43AmcmOSPJKcDFwP7+DknOAf4AuLCq7h9+mRvYSte/Ox8vaQRWDPeqegi4FLgeuAO4tqpuS3Jlkgt73X4LeAzwgSSfSbJ/ic21Z7nr352PlzQiK14KuV6auhRydrabYz90qBux79vXXf8+NdUF+kK7dsE995zoKiU1YNBLIQ339bRlSzdiXyiBhx8+8fVI2vSGeZ27jtdy8/HOxUtaR4b7elpqPv6CC5yLl7SuDPf1tNT9aK67zmvjJa0rw329TU93J08ffrh7nJ5e/tp4p2skDYHhPgpLzcU/4QlO10gaCsN9FJaai4flp2sc1UsakOE+CkvNxX/1q4v3n5+ucVQvaUCG+6gsNhe/3KWTy92gzBG9pAUM941kuVsZLHUSdn4Ev9SI3uCXxpLhvpEs91V+S43qt25dfkS/VPAb+lLTvP3AZjEf1P1BPjHxvcE+L+kOCIvd22bbNvjmN793WzMz3fPF7pMjaUPw9gOtWWpUv2uJ7yLfuXPpqZwHHlh8tH/ZZU7xSI0w3DeTxU7CLjdPv9RUzlKWCv3jneLxYCCNTlWN5OdZz3pWaUje856qXbuqku7xPe95pH1ioqqL4+5nYqJq27Zj21b6md/uYuu2bVv8NV73usXb+2tbrGZJywLmaoCMNdxbt1iIrjb0539/NQeErVuX3tZSrz9f21IHqqUOBh4oNEYMdy1vNaE/33c14X6i/grwQKExY7jr+Axrime5kfsw/wrYjAcKDyBaA8Ndw7ea0f5yITrMvwI224Fipd8Z1oHCg06zDHedOMcTPMP6K2CzHSiW+p2lDiDHc6AYl4POmE6vGe7a2Ib1V0ArB4qlfo7nQDEOB52NPL22zgcdw12b07D+w2zUA8VSv3MiDhQtHXQ26vTa8R50VsFwlzbigWKp31nuMtRRhuiJOFCMelujPujs2rWqt/VQwx3YA9wJHAAuX2T9o4D399Z/EphaaZuGuzaV9f5zfqWphFFNf2zUg85mm15bblvJqt6KQwt3YCtwF/CDwCnAzcBZC/r8S+BtvecXA+9fabuGu7TAqOd9N9NBZ6NOr22mkTvwY8D1fctvAt60oM/1wI/1np8EfIXeHSeX+jHcpU1iIx50jmdbJ2J6bTPNuQMvA97et/xK4PcW9LkV2NG3fBewfZFt7QXmgLmdO3euaockac024kFnlQYN9xXv557kZcCeqnpNb/mVwD+qqkv7+tza63O4t3xXr89Xltqu93OXpNUb5v3c7wNO71ve0WtbtE+Sk4DHAQ8MVqokadgGCfcbgTOTnJHkFLoTpvsX9NkP/Fzv+cuAP6uV/iSQJK2bk1bqUFUPJbmU7qTpVuDqqrotyZV0cz/7gT8C3p3kAPBVugOAJGlEVgx3gKq6DrhuQdub+55/C/jnwy1NknS8/Jo9SWrQilfLrNsLJ0eAgyt02053zfw4Gud9h/He/3Hedxjv/R9k33dV1eRKGxpZuA8iydwgl/y0aJz3HcZ7/8d532G893+Y++60jCQ1yHCXpAZt9HCfGXUBIzTO+w7jvf/jvO8w3vs/tH3f0HPukqTjs9FH7pKk42C4S1KDNmy4J9mT5M4kB5JcPup61lOSq5Pc37u75nzbE5J8JMnf9B4fP8oa10uS05PckOT2JLcluazXPi77/+gkn0pyc2//f73XfkaST/be/+/v3depSUm2Jvl0kv/WWx6nfb8nyWeTfCbJXK9tKO/9DRnuSbYCVwHnA2cBlyQ5a7RVrat30H2VYb/LgY9W1ZnAR3vLLXoIeGNVnQU8B/jF3r/1uOz/t4EXVtUzgbOBPUmeA7wV+A9V9cPA14BXj7DG9XYZcEff8jjtO8ALqursvuvbh/Le35DhDpwLHKiqu6vqQeAa4KIR17RuqupjdDdc63cR8M7e83cCP3VCizpBquqLVfVXveffoPtPfhrjs/9VVf+3t3hy76eAFwJ/3Gtvdv+T7AB+Enh7bzmMyb4vYyjv/Y0a7qcB9/YtH+61jZMnVdUXe8+/BDxplMWcCEmmgHPovmR9bPa/Ny3xGeB+4CN032T29ap6qNel5ff/7wD/Bni4t7yN8dl36A7k/zPJTUn29tqG8t4f6K6QGq2qqiRNX7Oa5DHAB4HXV9X/6QZwndb3v6q+A5yd5FTgT4B/MOKSTogkLwXur6qbkpw36npG5HlVdV+SJwIfSfLX/SvX8t7fqCP3Qb79qXVfTvJkgN7j/SOuZ90kOZku2Ger6kO95rHZ/3lV9XXgBrovpT+1961m0O77/7nAhUnuoZt6fSHwHxmPfQegqu7rPd5Pd2A/lyG99zdquA/y7U+t6/92q58D/ssIa1k3vTnWPwLuqKrf7ls1Lvs/2Ruxk+T7gJfQnXe4ge5bzaDR/a+qN1XVjqqaovs//mdVNc0Y7DtAku9P8tj558CPA7cypPf+hv2EapIL6Obj5r/9ad+IS1o3Sd4HnEd3u88vA28BPgxcC+ykuzXyy6tq4UnXTS/J84A/Bz7LI/Ouv0o37z4O+/8MupNmW+kGW9dW1ZVJfpBuNPsE4NPAK6rq26OrdH31pmX+VVW9dFz2vbeff9JbPAl4b1XtS7KNIbz3N2y4S5KO30adlpEkrYHhLkkNMtwlqUGGuyQ1yHCXpAYZ7pLUIMNdkhr0/wG1/SPCy/mzsQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "valeurs_n = np.arange(1, 50)\n",
    "valeurs_In = np.array([I(n) for n in valeurs_n])\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(valeurs_n, valeurs_In, 'ro')\n",
    "plt.title(\"Valeurs de $I_n$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- On conjecture que $I_n$ est décroissante. C'est évident puisque si on note $f_n(t)$ son intégrande, on observe que $f_{n+1}(t) \\leq f_n(t)$ pour tout $t$, et donc par monotonie de l'intégrale, $I_{n+1} \\leq I_n$.\n",
    "- On conjecture que $I_n \\to 0$. Cela se montre très facilement avec le théorème de convergence dominée."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5eafcdd8>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Valeurs de $\\\\ln(I_n)$ en fonction de $\\\\ln(n)$')"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(np.log(valeurs_n), np.log(valeurs_In), 'go')\n",
    "plt.title(r\"Valeurs de $\\ln(I_n)$ en fonction de $\\ln(n)$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Ce qu'on observe permet de conjecturer que $\\alpha=1$ est l'unique entier tel que $n^{\\alpha} I_n$ converge vers une limite non nulle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5ea83358>]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5eabfa90>]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f8c5ea83a58>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Valeurs de $n^{\\\\alpha} I_n$ et $n^{\\\\alpha} J_n$')"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "valeurs_Jn = np.array([J(n) for n in valeurs_n])\n",
    "alpha = 1\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(valeurs_n, valeurs_n**alpha * valeurs_In, 'r+', label=r'$n^{\\alpha} I_n$')\n",
    "plt.plot(valeurs_n, valeurs_n**alpha * valeurs_Jn, 'b+', label=r'$n^{\\alpha} J_n$')\n",
    "plt.legend()\n",
    "plt.title(r\"Valeurs de $n^{\\alpha} I_n$ et $n^{\\alpha} J_n$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- On en déduit qu'il en est de même pour $J_n$, on a $n^{\\alpha} J_n \\to l$ la même limite que $n^{\\alpha} I_n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Pour finir, on montre mathématiquement que $n^{\\alpha} (I_n - J_n)$ tend vers $0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5e7cdbe0>]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f8c5e9cb898>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Valeurs de $n^{\\\\alpha} (I_n - J_n)$')"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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3M0uQw93MLEEOdzOzBDnczcwSlCvcJS2V9JykXZJur9HutyWFpO7GlWhmZsNVN9wldQCrgCuBucCNkuYO0m4acCvwo0YXaWZmw5Nnz/1iYFdE7I6Io8DDwLJB2v0JcA9wpIH1mZnZCOQJ9zOBvRXTfdm8AZLeBZwVEf+tVkeSlkvqldTb398/7GLNzCyfUR9QlfQm4F7gE/XaRsTqiOiOiO7Ozs7RrrrpCusLrS7BzGxE8oT7PuCsiumubF7ZNGAesF7SHuASYG0KB1WLTxZbXYKZ2YjkCfcNwLmSzpE0GbgBWFteGBE/i4iZETE7ImYDPwSuiYjeplRsZmZ11Q33iDgG3AI8DuwEHo2IZyXdKemaZhc41grrC6goVBTAwGsP0ZjZRKKIaMmKu7u7o7d3fO/cqyiipzXfHzOzwUjaGBF1h719haqZWYIc7jX0LOppdQlmZiPicK+hsLjQ6hLMzEbE4W5mliCHu5lZghzuZmYJcribmSXI4W5mliCHu5lZghzuZmYJcribmSXI4W5mliCHu5lZghzuZmYJcribmSXI4W5mliCHu5lZghzuZmYJcribmSXI4W5mliCHu5lZghzuZmYJcribmSXI4T4ChfWFVpdgZlaTw30Eik8WW12CmVlNDnczswQ53HMqrC+golBRAAOvPURjZuORIqIlK+7u7o7e3t6WrHu0VBTR05rvm5m1N0kbI6K7XjvvuZuZJcjhPgI9i3paXYKZWU0O9xEoLC60ugQzs5oc7mZmCXK4m5klyOFuZpYgh7uZWYIc7mZmCXK4m5klKFe4S1oq6TlJuyTdPsjyfyFph6Stkr4r6ezGl2pmZnnVDXdJHcAq4EpgLnCjpLlVzTYB3RExH/gG8O8aXaiZmeWXZ8/9YmBXROyOiKPAw8CyygYRsS4iDmeTPwS6GlummZkNR55wPxPYWzHdl80bykeB/z7YAknLJfVK6u3v789fpZmZDUtDD6hKugnoBv50sOURsToiuiOiu7Ozs5GrNjOzCnnCfR9wVsV0VzbvBJKWAP8auCYiXm9MeROL7+1uZuNFnnDfAJwr6RxJk4EbgLWVDSRdANxPKdj3N77MicGP3zOz8aJuuEfEMeAW4HFgJ/BoRDwr6U5J12TN/hT4JeDrkjZLWjtEd2ZmNgb8JKZRKqwvDLrH3rOox7cGNrOGy/skJod7A/nxe2bWbH7MnplZG3O4N5Afv2dm44XDvYE8xm5m44XD3cwsQQ53M7MEOdzNzBLkcDczS5DD3cwsQQ53M7MEOdzHiO8YaWZjyeE+RnzHSDMbSw53M7MEOdybqLC+gIpCRQEMvPYQjZk1m+8KOUZ8x0gzawTfFdLMrI053MeI7xhpZmPJ4T5GfMdIMxtLDnczswQ53FvMZ86YWTM43FvMFzeZWTM43M3MEuRwbwFf3GRmzeaLmFrMFzeZ2XD4IiYzszbmcG+xoS5u8hCNmY2Gw73Fhrq4yWfRmNloONzNzBLkcB9HfBaNmTWKz5YZp2qdRVNYX/C9aszalM+WSZjH482sHof7OOVbBJvZaDjcx6nqYZc84/EemzezMo+5T0BDjcf7alez9HnM3QZ4j96s/TjcJ6DK8fg8wzW1DsA6+M3S5HCfgCrH4wuLC0RPDAzHlF/nPVVyqOAfKvRrfRj4g8Js/MgV7pKWSnpO0i5Jtw+y/M2SHsmW/0jS7EYXasMz2guihgr9Wn8FjMUHRTv31er1u6/m99VQEVHzC+gAngd+FZgMbAHmVrX558B/zF7fADxSr98LL7wwrLF61vUMOp8CJ7WjwElfle+vfk+9+SN5j/uaWOt3X83vKw+gN+rka0QwKUf+XwzsiojdAJIeBpYBOyraLAMK2etvAJ+XpKwQGyN5h2IKi39xhWvlGTblvf2y8utFZy/iyRefPGl+eey/co+9cpmvojVrnbqnQkr6ALA0Ij6WTX8Y+I2IuKWizfasTV82/XzW5kBVX8uB5QCzZs268MUXX2zkttgQat2uYLinVdY63bL6g2KwYZrqD4qywT4o6r2nHfpq9frdV/P7Gu6OUN5TIfMMy3wA+FLF9IeBz1e12Q50VUw/D8ys1a+HZcaHvEM59eaP5D3ua2Kt331NrGGZPAdU9wFnVUx3ZfMGbSNpEvAW4GCOvq3FhtpjGOr2B7Vui+BbJpiNHx2FQqFmg2Kx+DJQKBaLa4vF4mHgz4BPFwqF/oo2bwGuKBQKf1ksFq8HpkTE12v1u3r16sLy5ctHvQHWHItnLx7W/JEua+R72qGvVq/ffTW/r3qKxeJLhUJhdb12uW4/IOkq4LOUzpx5MCLuknQnpT8P1kqaAnwNuAD4O+CGyA7ADsW3HzAzG768Y+55zpYhIh4DHqua98cVr48A/2S4RZqZWXP4ClUzswQ53M3MEuRwNzNLkMPdzCxBLXtYh6R+oN4lqjOBA3XapKqdtx3ae/vbeduhvbc/z7afHRGd9TpqWbjnIak3zyk/KWrnbYf23v523nZo7+1v5LZ7WMbMLEEOdzOzBI33cK97iW3C2nnbob23v523Hdp7+xu27eN6zN3MzEZmvO+5m5nZCDjczcwSNC7Dvd4DuVMj6UFJ+7MnWpXnvVXSdyT9bfbvaa2ssVkknSVpnaQdkp6VdGs2v122f4qkpyVtyba/mM0/J3vY/K7s4fOTW11rs0jqkLRJ0l9m022x7ZL2SNomabOk3mxew37ux124S+oAVgFXAnOBGyXNbW1VTfdlYGnVvNuB70bEucB3s+kUHQM+ERFzgUuAj2f/3+2y/a8Dl0XEO4EFwFJJlwD3APdFxK8DrwAfbWGNzXYrsLNiup22/dKIWFBxbnvDfu7HXbhT8UDuiDgKlB/InayI+D6l++BXWgZ8JXv9FeDaMS1qjETESxHxTPb6EKVf8jNpn+2PiPi/2eQp2VcAl1F62DwkvP2SuoCrgS9l06JNtn0IDfu5H4/hfiawt2K6L5vXbk6PiJey1y8Dp7eymLEgaTalB778iDba/mxYYjOwH/gOpWcQvxoRx7ImKf8OfBb4V8DPs+kZtM+2B/A/JW2UVH4sXcN+7nM9rMNaKyJCUtLnrEr6JeCbwIqI+D+lHbiS1Lc/Io4DCyRNB74F/IMWlzQmJL0P2B8RGyUtbnU9LfDuiNgn6W3AdyT9TeXC0f7cj8c99zwP5G4HP5X0KwDZv/tbXE/TSDqFUrCviYi/yGa3zfaXRcSrwDrgHwLTs4fNQ7q/A78JXCNpD6Xh18uAz9Ee205E7Mv+3U/pQ/1iGvhzPx7DfQNwbnbEfDJwA7C2xTW1wlrgd7PXvwv8lxbW0jTZGOufAzsj4t6KRe2y/Z3ZHjuSpgLvpXTcYR3wgaxZktsfEZ+MiK6ImE3p9/x7EfE7tMG2SzpV0rTya+AKYDsN/Lkfl1eoDvZA7haX1FSS/jOwmNLtPn8K9ADfBh4FZlG6NfL1EVF90HXCk/Ru4AfANn4x7noHpXH3dtj++ZQOnHVQ2tl6NCLulPSrlPZm3wpsAm6KiNdbV2lzZcMy/zIi3tcO255t47eyyUnAQxFxl6QZNOjnflyGu5mZjc54HJYxM7NRcribmSXI4W5mliCHu5lZghzuZmYJcribmSXI4W5mlqD/D9vwUmfUInPhAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(valeurs_n, valeurs_n**alpha * (valeurs_In - valeurs_Jn), 'g+', label=r'$n^{\\alpha} (I_n - J_n)$')\n",
    "plt.legend()\n",
    "plt.title(r\"Valeurs de $n^{\\alpha} (I_n - J_n)$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Puis rapidement, on montre que $\\forall x \\geq 0, \\ln(1 + x) \\geq \\frac{x}{1+x}$. Ca peut se prouver de plein de façons différentes, mais par exemple on écrit $f(x) = (x+1) \\log(x+1) - x$ qui est de classe $\\mathcal{C}^1$, et on la dérive. $f'(x) = \\log(x+1) + 1 - 1 > 0$ donc $f$ est croissante, et $f(0) = 0$ donc $f(x) \\geq f(0) = 0$ pour tout $x \\geq 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5cf35978>]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5cf22080>]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f8c5cf220f0>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Comparaison entre deux fonctions')"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = np.linspace(0, 100, 10000)\n",
    "plt.plot(X, np.log(1 + X), 'r-', label=r'$\\log(1+x)$')\n",
    "plt.plot(X, X / (1 + X), 'b-', label=r'$\\frac{x}{1+x}$')\n",
    "plt.legend()\n",
    "plt.title(\"Comparaison entre deux fonctions\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## Planche 162"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On commence par définir la fonction, en utilisant `numpy.cos` et pas `math.cos` (les fonctions de `numpy` peuvent travailler sur des tableaux, c'est plus pratique)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x * (1 - x) * (1 + np.cos(5 * np.pi * x))\n",
    "\n",
    "Xs = np.linspace(0, 1, 2000)\n",
    "Ys = f(Xs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pas besoin de lire le maximum sur un graphique :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sur [0, 1], avec 2000 points, M = 0.4812601808328304\n"
     ]
    }
   ],
   "source": [
    "M = max_de_f = max(Ys)\n",
    "print(\"Sur [0, 1], avec 2000 points, M =\", M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On affiche la fonction, comme demandé, avec un titre :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0c69f16748>]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Fonction $f(x)$ sur $[0,1]$')"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(Xs, Ys)\n",
    "plt.title(\"Fonction $f(x)$ sur $[0,1]$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pour calculer l'intégrale, on utilise `scipy.integrate.quad` :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def In(x, n):\n",
    "    def fn(x):\n",
    "        return f(x) ** n\n",
    "    return integrate.quad(fn, 0, 1)[0]\n",
    "\n",
    "def Sn(x):\n",
    "    return np.sum([In(Xs, n) * x**n for n in range(0, n+1)], axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On vérifie avant de se lancer dans l'affichage :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "In(x, 0 ) = 1.0\n",
      "In(x, 1 ) = 0.16666666666666663\n",
      "In(x, 2 ) = 0.049987680821294386\n",
      "In(x, 3 ) = 0.017874498250810132\n",
      "In(x, 4 ) = 0.0069477925896280456\n",
      "In(x, 5 ) = 0.0028398023087289567\n",
      "In(x, 6 ) = 0.0012011683591199485\n",
      "In(x, 7 ) = 0.00052074827894252\n",
      "In(x, 8 ) = 0.00022991112607879838\n",
      "In(x, 9 ) = 0.00010290185158968158\n"
     ]
    }
   ],
   "source": [
    "for n in range(10):\n",
    "    print(\"In(x,\", n, \") =\", In(Xs, n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0c69cda3c8>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0c69cda860>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0c69cdaac8>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0c69cdaf28>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0c69ce36a0>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f0c69d18550>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = 1/M + 0.1\n",
    "X2s = np.linspace(-a, a, 2000)\n",
    "\n",
    "plt.figure()\n",
    "for n in [10, 20, 30, 40, 50]:\n",
    "    plt.plot(X2s, Sn(X2s), label=\"n =\" + str(n))\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$S_n(x)$ semble diverger pour $x\\to2^-$ quand $n\\to\\infty$.\n",
    "Le rayon de convergence de la série $\\sum In x^n$ **semble** être $2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def un(n):\n",
    "    return In(Xs, n + 1) / In(Xs, n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "un = 0.16666666666666663 pour n = 0\n",
      "un = 0.2999260849277664 pour n = 1\n",
      "un = 0.35757806637822104 pour n = 2\n",
      "un = 0.38869860804697826 pour n = 3\n",
      "un = 0.4087344681198935 pour n = 4\n",
      "un = 0.4229760485185215 pour n = 5\n",
      "un = 0.43353479550864377 pour n = 6\n",
      "un = 0.44150146121592054 pour n = 7\n",
      "un = 0.4475723004132197 pour n = 8\n",
      "un = 0.4522404636909894 pour n = 9\n"
     ]
    }
   ],
   "source": [
    "for n in range(10):\n",
    "    print(\"un =\", un(n), \"pour n =\", n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ici, `un` ne peut pas être utilisé comme une fonction \"numpy\" qui travaille sur un tableau, on stocke donc les valeurs \"plus manuellement\" :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "def affiche_termes_un(N):\n",
    "    valeurs_un = [0] * N\n",
    "    for n in range(N):\n",
    "        valeurs_un[n] = un(n)\n",
    "\n",
    "    plt.figure()\n",
    "    plt.plot(valeurs_un, 'o-')\n",
    "    plt.title(\"Suite $u_n$\")\n",
    "    plt.grid()\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "affiche_termes_un(30)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La suite $u_n$ semble être croissante (on peut le prouver), toujours plus petite que $1$ (se prouve facilement aussi, $I_{n+1} < I_n$), et semble converger.\n",
    "Peut-être vers $1/2$, il faut aller regarder plus loin ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "affiche_termes_un(100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pour conclure, on peut prouver que la suite est monotone et bornée, donc elle converge.\n",
    "Il est plus dur de calculer sa limite, et cela sort de l'exercice."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## Planche 166"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "case_max = 12\n",
    "univers = list(range(case_max))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "def prochaine_case(case):\n",
    "    return (case + rd.randint(1, 6+1)) % case_max"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Yn(duree, depart=0):\n",
    "    case = depart\n",
    "    for coup in range(duree):\n",
    "        case = prochaine_case(case)\n",
    "    return case"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Avant de s'en servir pour simuler plein de trajectoirs, on peut vérifier :\n",
    "\n",
    "- en un coup, on avance pas plus de 6 cases :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 3, 3, 3, 6, 6, 4, 1, 2, 2]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[Yn(1) for _ in range(10)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- En 100 coups, on commence à ne plus voir de tendance :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 9, 2, 5, 2, 2, 0, 4, 4]"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[Yn(100) for _ in range(10)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pour l'histogramme, on triche un peu en utilisant `numpy.bincount`. Mais on peut le faire à la main très simplement !"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3, 0, 2, 1, 0, 0, 0, 1, 0, 0])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.bincount(_, minlength=case_max)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "def histogramme(duree, repetitions=5000):\n",
    "    cases = [Yn(duree) for _ in range(repetitions)]\n",
    "    frequences = np.bincount(cases, minlength=case_max)\n",
    "    # aussi a la main si besoin\n",
    "    frequences = [0] * case_max\n",
    "    for case in cases:\n",
    "        frequences[case] += 1\n",
    "    return frequences / np.sum(frequences)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.085 , 0.0762, 0.0874, 0.0818, 0.09  , 0.0818, 0.0854, 0.086 ,\n",
       "       0.079 , 0.0844, 0.0826, 0.0804])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "histogramme(50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "def voir_histogramme(valeurs_n):\n",
    "    for n in valeurs_n:\n",
    "        plt.figure()\n",
    "        plt.bar(np.arange(case_max), histogramme(n))\n",
    "        plt.title(\"Histogramme de cases visitées en \" + str(n) + \" coups\")\n",
    "        plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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CUrek2Q3WHyrpDknrJR1XWj5Z0u8kLZN0l6TjS+uukPSApKXpNrk1XTIzs4Ea3qyApGHARcARwBpgiaTOiLi3VOwh4FTgs3WbvwicHBH3SdoLuF3S4oh4Oq3/XEQsHGgnzMystZqGAzAV6I6IlQCS5gPTgT+HQ0SsSus2lDeMiD+Uph+W9BjQATyNmZltsaoMK40GVpfm16RlfSJpKjACuL+0+Lw03HShpJF9rdPMzDaPQTkhLWlP4CrgIxFRO7o4G9gXOAjYHTirl23PkNQlqaunp2cwmmtm9rpXJRzWAmNL82PSskok7Qz8GPhiRNxaWx4R66LwCnA5xfBVJiLmRsSUiJjS0dFRdbdmZjYAVcJhCTBR0nhJI4BZQGeVylP564Ef1J94TkcTSBIwA7inLw03M7PNp2k4RMR64ExgMbAcWBARyyTNkXQsgKSDJK0BZgKXSFqWNv8QcChwaoOPrF4j6W7gbmAU8LWW9szMzPqtyqeViIhFwKK6ZeeUppdQDDfVb3c1cHUvdR7Wp5aamdmg8Tekzcws43AwM7OMw8HMzDIOBzMzyzgczMws43AwM7OMw8HMzDIOBzMzyzgczMws43AwM7OMw8HMzDIOBzMzyzgczMws43AwM7OMw8HMzDIOBzMzyzgczMws43AwM7OMw8HMzDIOBzMzyzgczMwsUykcJE2TtEJSt6TZDdYfKukOSeslHVe37hRJ96XbKaXlB0q6O9X5bUkaeHfMzKwVmoaDpGHARcBRwCTgBEmT6oo9BJwKzKvbdnfgK8DBwFTgK5J2S6u/C5wOTEy3af3uhZmZtVSVI4epQHdErIyIV4H5wPRygYhYFRF3ARvqtj0SuDkinoyIp4CbgWmS9gR2johbIyKAHwAzBtoZMzNrjSrhMBpYXZpfk5ZV0du2o9N00zolnSGpS1JXT09Pxd2amdlAbPEnpCNibkRMiYgpHR0d7W6OmdnrQpVwWAuMLc2PScuq6G3btWm6P3WamdlmViUclgATJY2XNAKYBXRWrH8x8AFJu6UT0R8AFkfEOuBZSYekTymdDNzQj/abmdlm0DQcImI9cCbFC/1yYEFELJM0R9KxAJIOkrQGmAlcImlZ2vZJ4KsUAbMEmJOWAXwC+D7QDdwP3NTSnpmZWb8Nr1IoIhYBi+qWnVOaXsLGw0TlcpcBlzVY3gXs35fGmpnZ4NjiT0ibmdngcziYmVnG4WBmZhmHg5mZZRwOZmaWcTiYmVnG4WBmZhmHg5mZZRwOZmaWcTiYmVnG4WBmZhmHg5mZZRwOZmaWcTiYmVnG4WBmZhmHg5mZZRwOZmaWcTiYmVnG4WBmZhmHg5mZZRwOZmaWqRQOkqZJWiGpW9LsButHSrourb9N0ri0/ERJS0u3DZImp3W/THXW1r2xlR0zM7P+axoOkoYBFwFHAZOAEyRNqit2GvBUREwALgQuAIiIayJickRMBk4CHoiIpaXtTqytj4jHWtAfMzNrgSpHDlOB7ohYGRGvAvOB6XVlpgNXpumFwOGSVFfmhLStmZlt4aqEw2hgdWl+TVrWsExErAeeAfaoK3M8cG3dssvTkNKXG4QJAJLOkNQlqaunp6dCc83MbKAG5YS0pIOBFyPintLiEyPiHcB70+2kRttGxNyImBIRUzo6OgahtWZmViUc1gJjS/Nj0rKGZSQNB3YBniitn0XdUUNErE1/nwPmUQxfmZnZFqBKOCwBJkoaL2kExQt9Z12ZTuCUNH0ccEtEBICkbYAPUTrfIGm4pFFpelvgGOAezMxsizC8WYGIWC/pTGAxMAy4LCKWSZoDdEVEJ3ApcJWkbuBJigCpORRYHRErS8tGAotTMAwDfgZ8ryU9MjOzAWsaDgARsQhYVLfsnNL0y8DMXrb9JXBI3bIXgAP72FYzMxsk/oa0mZllHA5mZpZxOJiZWcbhYGZmGYeDmZllHA5mZpZxOJiZWcbhYGZmGYeDmZllHA5mZpZxOJiZWcbhYGZmGYeDmZllHA5mZpZxOJiZWcbhYGZmGYeDmZllHA5mZpZxOJiZWcbhYGZmGYeDmZllKoWDpGmSVkjqljS7wfqRkq5L62+TNC4tHyfpJUlL0+3i0jYHSro7bfNtSWpVp8zMbGCahoOkYcBFwFHAJOAESZPqip0GPBURE4ALgQtK6+6PiMnp9vHS8u8CpwMT021a/7thZmatVOXIYSrQHRErI+JVYD4wva7MdODKNL0QOHxTRwKS9gR2johbIyKAHwAz+tx6MzPbLKqEw2hgdWl+TVrWsExErAeeAfZI68ZLulPSryS9t1R+TZM6AZB0hqQuSV09PT0VmmtmZgO1uU9IrwPeHBEHAJ8B5knauS8VRMTciJgSEVM6Ojo2SyPNzGxjVcJhLTC2ND8mLWtYRtJwYBfgiYh4JSKeAIiI24H7gbel8mOa1GlmZm1SJRyWABMljZc0ApgFdNaV6QROSdPHAbdEREjqSCe0kfQWihPPKyNiHfCspEPSuYmTgRta0B8zM2uB4c0KRMR6SWcCi4FhwGURsUzSHKArIjqBS4GrJHUDT1IECMChwBxJfwQ2AB+PiCfTuk8AVwDbAzelm5mZbQGahgNARCwCFtUtO6c0/TIws8F2PwR+2EudXcD+fWmsmZkNDn9D2szMMg4HMzPLOBzMzCzjcDAzs4zDwczMMg4HMzPLOBzMzCzjcDAzs4zDwczMMg4HMzPLOBzMzCzjcDAzs4zDwczMMg4HMzPLOBzMzCzjcDAzs4zDwczMMg4HMzPLOBzMzCzjcDAzs4zDwczMMpXCQdI0SSskdUua3WD9SEnXpfW3SRqXlh8h6XZJd6e/h5W2+WWqc2m6vbFVnTIzs4EZ3qyApGHARcARwBpgiaTOiLi3VOw04KmImCBpFnABcDzwOPDBiHhY0v7AYmB0absTI6KrRX0xM7MWqXLkMBXojoiVEfEqMB+YXldmOnBlml4IHC5JEXFnRDycli8Dtpc0shUNNzOzzadKOIwGVpfm17Dxu/+NykTEeuAZYI+6Mn8D3BERr5SWXZ6GlL4sSY12LukMSV2Sunp6eio018zMBmpQTkhL2o9iqOljpcUnRsQ7gPem20mNto2IuRExJSKmdHR0bP7GmplZpXBYC4wtzY9JyxqWkTQc2AV4Is2PAa4HTo6I+2sbRMTa9Pc5YB7F8JWZmW0BqoTDEmCipPGSRgCzgM66Mp3AKWn6OOCWiAhJuwI/BmZHxG9qhSUNlzQqTW8LHAPcM7CumJlZqzQNh3QO4UyKTxotBxZExDJJcyQdm4pdCuwhqRv4DFD7uOuZwATgnLqPrI4EFku6C1hKceTxvVZ2zMzM+q/pR1kBImIRsKhu2Tml6ZeBmQ22+xrwtV6qPbB6M83MbDD5G9JmZpZxOJiZWcbhYGZmGYeDmZllHA5mZpZxOJiZWcbhYGZmGYeDmZllHA5mZpZxOJiZWcbhYGZmGYeDmZllHA5mZpZxOJiZWcbhYGZmGYeDmZllHA5mZpZxOJiZWcbhYGZmGYeDmZllHA5mZpapFA6SpklaIalb0uwG60dKui6tv03SuNK6s9PyFZKOrFqnmZm1T9NwkDQMuAg4CpgEnCBpUl2x04CnImICcCFwQdp2EjAL2A+YBvyrpGEV6zQzszapcuQwFeiOiJUR8SowH5heV2Y6cGWaXggcLklp+fyIeCUiHgC6U31V6jQzszYZXqHMaGB1aX4NcHBvZSJivaRngD3S8lvrth2dppvVCYCkM4Az0uzzklZUaPNAjAIe38z7qEwXDLiKSv1pwX4qcX82skX1pQW2qOfOAG1NfQHYp68bVAmHtoqIucDcwdqfpK6ImDJY+9vc3J8t19bUF9i6+rM19QWK/vR1myrDSmuBsaX5MWlZwzKShgO7AE9sYtsqdZqZWZtUCYclwERJ4yWNoDjB3FlXphM4JU0fB9wSEZGWz0qfZhoPTAT+s2KdZmbWJk2HldI5hDOBxcAw4LKIWCZpDtAVEZ3ApcBVkrqBJyle7EnlFgD3AuuBT0bEnwAa1dn67vXLoA1hDRL3Z8u1NfUFtq7+bE19gX70R8UbfDMzs9f4G9JmZpZxOJiZWcbhULI1XdJD0lhJv5B0r6Rlkj7V7jYNVPp2/Z2Sbmx3WwZK0q6SFkr6vaTlkt7V7jb1l6R/SI+xeyRdK2m7drepLyRdJukxSfeUlu0u6WZJ96W/u7WzjX3RS3/+OT3W7pJ0vaRdm9XjcEi2wkt6rAf+MSImAYcAnxzi/QH4FLC83Y1okW8BP4mIfYH/zhDtl6TRwP8CpkTE/hQfMJnV3lb12RUUl/cpmw38PCImAj9P80PFFeT9uRnYPyL+G/AH4OxmlTgcXrNVXdIjItZFxB1p+jmKF5/Rm95qyyVpDPDXwPfb3ZaBkrQLcCjFp/yIiFcj4un2tmpAhgPbp+847QA83Ob29ElE/JriU5Zl5UsCXQnMGNRGDUCj/kTETyNifZq9leK7ZZvkcHhNo8uEDNkX07J0ldwDgNva25IB+SbweWBDuxvSAuOBHuDyNEz2fUlvaHej+iMi1gLfAB4C1gHPRMRP29uqlnhTRKxL048Ab2pnY1rso8BNzQo5HLZyknYEfgh8OiKebXd7+kPSMcBjEXF7u9vSIsOBdwLfjYgDgBcYWsMWf5bG4qdTBN5ewBskfbi9rWqt9IXereIz/5K+SDHkfE2zsg6H12x1l/SQtC1FMFwTET9qd3sG4N3AsZJWUQz3HSbp6vY2aUDWAGsionYkt5AiLIai9wMPRERPRPwR+BHwl21uUys8KmlPgPT3sTa3Z8AknQocA5wYFb7g5nB4zVZ1SY90yfRLgeUR8S/tbs9ARMTZETEmIsZR/F9uiYgh++40Ih4BVkuqXSnzcIqrCAxFDwGHSNohPeYOZ4ieXK9TviTQKcANbWzLgEmaRjEse2xEvFhlG4dDkk7W1C7psRxYsAVd0qM/3g2cRPEue2m6Hd3uRtmf/T1wjaS7gMnA19vcnn5JRz8LgTuAuyleU4bUpSckXQv8DthH0hpJpwHnA0dIuo/i6Oj8draxL3rpz3eAnYCb02vBxU3r8eUzzMysno8czMws43AwM7OMw8HMzDIOBzMzyzgczMws43AwM7OMw8HMzDL/H2l5sXqkX7liAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "voir_histogramme([1, 2, 3, 50, 100, 200])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On s'approche d'une distribution uniforme !"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On a tout simplement l'expression suivante :\n",
    "$$\\forall n \\geq 0, \\mathbb{P}(Y_{n+1} = k) = \\frac{1}{6} \\sum_{\\delta = 1}^{6} \\mathbb{P}(Y_n = k - \\delta \\mod 12).$$\n",
    "Avec $k - 1 \\mod 12 = 11$ si $k = 0$ par exemple. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On a donc la matrice suivante pour exprimer $U_n = (\\mathbb{P}(Y_n = k))_{0\\leq k \\leq 11}$ en fonction de $U_{n-1}$ :\n",
    "\n",
    "$$ P = \\frac{1}{6} \\begin{bmatrix}\n",
    "0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n",
    "1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1\\\\\n",
    "1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1\\\\\n",
    "1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1\\\\\n",
    "1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\\\\n",
    "1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n",
    "1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\\\\n",
    "\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On va la définir rapidement en Python, et calculer ses valeurs propres notamment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = np.zeros((case_max, case_max))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "for k in range(case_max):\n",
    "    for i in range(k - 6, k):\n",
    "        P[k, i] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0., 0., 0., 0., 0., 0., 1., 1., 1., 1., 1., 1.],\n",
       "       [1., 0., 0., 0., 0., 0., 0., 1., 1., 1., 1., 1.],\n",
       "       [1., 1., 0., 0., 0., 0., 0., 0., 1., 1., 1., 1.],\n",
       "       [1., 1., 1., 0., 0., 0., 0., 0., 0., 1., 1., 1.],\n",
       "       [1., 1., 1., 1., 0., 0., 0., 0., 0., 0., 1., 1.],\n",
       "       [1., 1., 1., 1., 1., 0., 0., 0., 0., 0., 0., 1.],\n",
       "       [1., 1., 1., 1., 1., 1., 0., 0., 0., 0., 0., 0.],\n",
       "       [0., 1., 1., 1., 1., 1., 1., 0., 0., 0., 0., 0.],\n",
       "       [0., 0., 1., 1., 1., 1., 1., 1., 0., 0., 0., 0.],\n",
       "       [0., 0., 0., 1., 1., 1., 1., 1., 1., 0., 0., 0.],\n",
       "       [0., 0., 0., 0., 1., 1., 1., 1., 1., 1., 0., 0.],\n",
       "       [0., 0., 0., 0., 0., 1., 1., 1., 1., 1., 1., 0.]])"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy.linalg as LA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [],
   "source": [
    "spectre, vecteur_propres = LA.eig(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On a besoin d'éliminer les erreurs d'arrondis, mais on voit que $6$ est valeur propre, associée au vecteur $[0,\\dots,1,\\dots,0]$ avec un $1$ seulement à la 8ème composante."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 6.+0.j, -1.+4.j, -1.-4.j, -1.+1.j, -1.-1.j, -1.+0.j, -1.-0.j,\n",
       "        0.+0.j,  0.+0.j,  0.-0.j, -0.+0.j, -0.-0.j])"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.round(spectre)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.+0.j,  0.+0.j,  0.-0.j, -0.-0.j, -0.+0.j,  0.-0.j,  0.+0.j,\n",
       "        1.+0.j, -0.-0.j, -0.+0.j,  0.+0.j,  0.-0.j])"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.round(vecteur_propres[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$P$ n'est pas diagonalisable, **à prouver** au tableau si l'examinateur le demande."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Planche 168"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Soit $f(x) = \\frac{1}{2 - \\exp(x)}$, et $a(n) = \\frac{f^{(n)}(0)}{n!}$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return 1 / (2 - np.exp(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Soit $g(x) = 2 - \\exp(x)$, telle que $g(x) f(x) = 1$. En dérivant $n > 0$ fois cette identité et en utilisant la formule de Leibniz, on trouve :\n",
    "$$ (g(x)f(x))^{(n)} = 0 = \\sum_{k=0}^n {n \\choose k} g^{(k)}(x) f^{(n-k)}(x).$$\n",
    "Donc en $x=0$, on utilise que $g^{(k)}(x) = - \\exp(x)$, qui donne que $g^{(k)}(0) = 1$ si $k=0$ ou $-1$ sinon, pour trouver que $\\sum_{k=0}^n {n \\choose k} f^{(k)}(0) = f^{(n)}(0)$. En écrivant ${n \\choose k} = \\frac{k! (n-k)!}{n!}$ et avec la formule définissant $a(n)$, cela donne directement la somme recherchée : $$ a(n) = \\sum_{k=1}^n \\frac{a(n-k)}{k!}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Pour calculer $a(n)$ avec Python, on utilise cette formule comme une formule récursive, et on triche un peu en utilisant `math.factorial` pour calculer $k!$. Il nous faut aussi $a(0) = f(0) = 1$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import factorial\n",
    "\n",
    "def a_0an(nMax):\n",
    "    valeurs_a = np.zeros(nMax+1)\n",
    "    valeurs_a[0] = 1.0\n",
    "    for n in range(1, nMax+1):\n",
    "        valeurs_a[n] = sum(valeurs_a[n-k] / factorial(k) for k in range(1, n+1))\n",
    "    return valeurs_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 0 on a a(n) = 1.0\n",
      "Pour n = 1 on a a(n) = 1.0\n",
      "Pour n = 2 on a a(n) = 1.5\n",
      "Pour n = 3 on a a(n) = 2.1666666666666665\n",
      "Pour n = 4 on a a(n) = 3.1249999999999996\n",
      "Pour n = 5 on a a(n) = 4.508333333333334\n",
      "Pour n = 6 on a a(n) = 6.504166666666667\n",
      "Pour n = 7 on a a(n) = 9.383531746031748\n",
      "Pour n = 8 on a a(n) = 13.537574404761907\n",
      "Pour n = 9 on a a(n) = 19.530591380070547\n",
      "Pour n = 10 on a a(n) = 28.176687334656087\n"
     ]
    }
   ],
   "source": [
    "nMax = 10\n",
    "valeurs_n = np.arange(0, nMax + 1)\n",
    "valeurs_a = a_0an(nMax)\n",
    "\n",
    "for n in valeurs_n:\n",
    "    print(\"Pour n =\", n, \"on a a(n) =\", valeurs_a[n])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5cc63710>]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5cc95ba8>]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5cc63da0>]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'$a(n)$ et deux autres suites')"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f8c5cccc080>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(valeurs_n, valeurs_a, 'ro', label=r'$a(n)$')\n",
    "plt.plot(valeurs_n, 1 / np.log(2)**valeurs_n, 'g+', label=r'$1/\\log(2)^n$')\n",
    "plt.plot(valeurs_n, 1 / (2 * np.log(2)**valeurs_n), 'bd', label=r'$1/(2\\log(2)^n)$')\n",
    "plt.title(\"$a(n)$ et deux autres suites\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- On observe que $a(n)$ est comprise entre $\\frac{1}{2(\\log(2))^n}$ et $\\frac{1}{\\log(2)^n}$, donc le rayon de convergence de $S(x) = \\sum a(n) x^n$ est $\\log(2)$.\n",
    "- On va calculer les sommes partielles $S_n(x)$ de la série $S(x)$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Sn(x, n):\n",
    "    valeurs_a = a_0an(n)\n",
    "    return sum(valeurs_a[k] * x**k for k in range(0, n + 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut vérifie que notre fonction marche :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour n = 0 S_n(x) = 1.0\n",
      "Pour n = 1 S_n(x) = 1.5\n",
      "Pour n = 2 S_n(x) = 1.875\n",
      "Pour n = 3 S_n(x) = 2.1458333333333335\n",
      "Pour n = 4 S_n(x) = 2.3411458333333335\n",
      "Pour n = 5 S_n(x) = 2.4820312500000004\n",
      "Pour n = 6 S_n(x) = 2.583658854166667\n"
     ]
    }
   ],
   "source": [
    "x = 0.5\n",
    "for n in range(0, 6 + 1):\n",
    "    print(\"Pour n =\", n, \"S_n(x) =\", Sn(x, n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "valeurs_x = np.linspace(0, 0.5, 1000)\n",
    "valeurs_f = f(valeurs_x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"color:red;\">Je pense que l'énoncé comporte une typo sur l'intervale ! Vu le rayon de convergence, on ne voit rien si on affiche sur $[0,10]$ !</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5c49f438>]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5c4ebac8>]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5c4ebf28>]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5c4eb2e8>]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5c5035f8>]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5c542940>]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5c5429e8>]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8c5c310400>]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'$f(x)$ et $S_n(x)$ pour $n = 0$ à $n = 6$')"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f8c5c3dda20>"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "for n in range(0, 6 + 1):\n",
    "    valeurs_Sn = []\n",
    "    for x in valeurs_x:\n",
    "        valeurs_Sn.append(Sn(x, n))\n",
    "    plt.plot(valeurs_x, valeurs_Sn, ':', label='$S_' + str(n) + '(x)$')\n",
    "plt.plot(valeurs_x, valeurs_f, '-', label='$f(x)$')\n",
    "plt.title(\"$f(x)$ et $S_n(x)$ pour $n = 0$ à $n = 6$\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Planche 170"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def u(n):\n",
    "    return np.arctan(n+1) - np.arctan(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fa9b5ac27f0>]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Premières valeurs de $u_n$')"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "valeurs_n = np.arange(50)\n",
    "valeurs_u = u(valeurs_n)\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(valeurs_n, valeurs_u, \"o-\")\n",
    "plt.title(\"Premières valeurs de $u_n$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut vérifier le prognostic quand à la somme de la série $\\sum u_n$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5707963267948966"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.550798992821746"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(valeurs_u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.01273069820194558"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "somme_serie = pi/2\n",
    "somme_partielle = sum(valeurs_u)\n",
    "erreur_relative = abs(somme_partielle - somme_serie) / somme_serie\n",
    "erreur_relative"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Avec seulement $50$ termes, on a moins de $1.5%$ d'erreur relative, c'est déjà pas mal !"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$(u_n)_n$ semble être décroisante, et tendre vers $0$. On peut prouver ça mathématiquement."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On sait aussi que $\\forall x\\neq0, \\arctan(x) + \\arctan(1/x) = \\frac{\\pi}{2}$, et que $\\arctan(x) \\sim x$, donc on obtient que $u_n \\sim \\frac{1}{n} - \\frac{1}{n+1} = \\frac{1}{n(n+1)}$.\n",
    "On peut le vérifier :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fa9b5ba2e10>]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Valeurs de $u_n / \\\\frac{1}{n(n+1)}$')"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "valeurs_n = np.arange(10, 1000)\n",
    "valeurs_u = u(valeurs_n)\n",
    "valeurs_equivalents = 1 / (valeurs_n * (valeurs_n + 1))\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(valeurs_n, valeurs_u / valeurs_equivalents, \"-\")\n",
    "plt.title(r\"Valeurs de $u_n / \\frac{1}{n(n+1)}$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Pour $e = (e_n)_{n\\in\\mathbb{N}}$ une suite de nombres égaux à $0$ ou $1$ (*i.e.*, $\\forall n, e_n \\in \\{0,1\\}$, $S_n(e) = \\sum{i=0}^n e_i u_i$ est bornée entre $0$ et $\\sum_{i=0}^n u_i$. Et $u_n \\sim \\frac{1}{n(n+1)}$ qui est le terme général d'une série convergente (par critère de Cauchy, par exemple, avec $\\alpha=2$). Donc la série $\\sum u_n$ converge et donc par encadrement, $S_n(e)$ converge pour $n\\to\\infty$, *i.e.*, $S(e)$ converge. Ces justifications donnent aussi que $$0 \\leq S(e) \\leq \\sum_{n\\geq0} u_n = \\lim_{n\\to\\infty} \\arctan(n) - \\arctan(0) = \\frac{\\pi}{2}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Pour $e = (0, 1, 0, 1, \\ldots)$, $S(e)$ peut être calculée avec Python. Pour trouver une valeur approchée à $\\delta = 10^{-5}$ près, il faut borner le **reste** de la série, $R_n(e) = \\sum_{i \\geq n + 1} e_i u_i$. Ici, $R_{2n+1}(e) \\leq u_{2n+2}$ or $u_i \\leq \\frac{1}{i(i+1)}$, donc $R_{2n+1}(e) \\leq \\frac{1}{(2n+1)(2n+2)}$. $\\frac{1}{(2n+1)(2n+2)} \\leq \\delta$ dès que $2n+1 \\geq \\sqrt{\\delta}$, *i.e.*, $n \\geq \\frac{\\sqrt{\\delta}+1}{2}$. Calculons ça :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import ceil, sqrt, pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Se(e, delta=1e-5, borne_sur_n_0=10000):\n",
    "    borne_sur_n_1 = int(ceil(1 + sqrt(delta)/2.0))\n",
    "    borne_sur_n = max(borne_sur_n_0, borne_sur_n_1)\n",
    "    somme_partielle = 0\n",
    "    for n in range(0, borne_sur_n + 1):\n",
    "        somme_partielle += e(n) * u(n)\n",
    "    return somme_partielle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "def e010101(n):\n",
    "    return 1 if n % 2 == 0 else 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour delta = 1e-05 on a Se010101(delta) ~= 1.06408\n"
     ]
    }
   ],
   "source": [
    "delta = 1e-5\n",
    "Se010101 = Se(e010101, delta)\n",
    "print(\"Pour delta =\", delta, \"on a Se010101(delta) ~=\", round(Se010101, 5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Pour inverser la fonction, et trouver la suite $e$ telle que $S(e) = x$ pour un $x$ donné, il faut réfléchir un peu plus."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "def inverse_Se(x, n):\n",
    "    assert 0 < x < pi/2.0, \"Erreur : x doit être entre 0 et pi/2 strictement.\"\n",
    "    print(\"Je vous laisse chercher.\")\n",
    "    raise NotImplementedError"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ca suffit pour la partie Python."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Planche 172"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "from random import random\n",
    "\n",
    "def pile(proba):\n",
    "    \"\"\" True si pile, False si face (false, face, facile à retenir).\"\"\"\n",
    "    return random() < proba"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- D'abord, on écrit une fonction pour **simuler** l'événement aléatoire :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "def En(n, p):\n",
    "    lance = pile(p)\n",
    "    for i in range(n - 1):\n",
    "        nouveau_lance = pile(p)\n",
    "        if lance and nouveau_lance:\n",
    "            return False\n",
    "        nouveau_lance = lance\n",
    "    return True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([24, 76])"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lances = [ En(2, 0.5) for _ in range(100) ]\n",
    "np.bincount(lances)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "def pn(n, p, nbSimulations=100000):\n",
    "    return np.mean([ En(n, p) for _ in range(nbSimulations) ])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Par exemple, pour seulement $2$ lancés, on a $1 - p_n = p^2$ car $\\overline{E_n}$ est l'événement d'obtenir $2$ piles qui est de probabilité $p^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.75281"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pn(2, 0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Avec $4$ lancés, on a $p_n$ bien plus petit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.56108"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pn(4, 0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- On vérifie que $p_n(n, p)$ est décroissante en $p$, à $n$ fixé :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.97252"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pn(4, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10009"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pn(4, 0.9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- On vérifie que $p_n(n, p)$ est décroissante en $n$, à $p$ fixé :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.86586"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pn(6, 0.2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.80334"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pn(20, 0.2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.80021"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pn(100, 0.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Notons que la suite semble converger ? Ou alors elle décroit de moins en moins rapidement."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Par récurrence et en considérant les possibles valeurs des deux derniers lancés numérotés $n+2$ et $n+1$, on peut montrer que\n",
    "  $$\\forall n, p_{n+2} = (1-p) p_{n+1} + p(1-p) p_n$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Si $p_n$ converge, on trouve sa limite $l$ comme point fixe de l'équation précédente. $l = (1-p) l + p(1-p) l$ ssi $1 = 1-p + p(1-p)$ ou $l=0$, donc si $p\\neq0$, $l=0$. Ainsi l'événement \"on obtient deux piles d'affilé sur un nombre infini de lancers$ est bien presque sûr."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Je vous laisse terminer pour calculer $T$ et les dernières questions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Planche 177\n",
    "\n",
    "- Le domaine de définition de $f(x) = \\sum_{n \\geq 1} \\frac{x^n}{n^2}$ est $[-1, 1]$ car $\\sum \\frac{x^n}{n^k}$ converge si $\\sum x^n$ converge (par $k$ dérivations successives), qui converge ssi $|x| < 1$. Et en $-1$ et $1$, on utilise $\\sum \\frac{1}{n^2} = \\frac{\\pi^2}{6}$.\n",
    "\n",
    "- Pour calculer $f(x)$ à $10^{-5}$ près, il faut calculer sa somme partielle $S_n(x) := \\sum_{i=1}^n \\frac{x^i}{i^2}$ en bornant son reste $S_n(x) := \\sum_{i \\geq n+1} \\frac{x^i}{i^2}$ par (au moins) $10^{-5}$. Une inégalité montre rapidement que $R_n(x) \\leq |x|^{n+1}\\sum_{i\\geq n+1} \\frac{1}{i^2} $, et donc $R_n(x) \\leq \\delta$ dès que $|x|^{n+1} \\leq \\frac{\\pi^2}{6} \\delta$, puisque $\\sum_{i\\geq n+1} \\frac{1}{i^2} \\leq \\sum_{i=0}^{+\\infty} \\frac{1}{i^2} = \\frac{\\pi^2}{6}$. En inversant pour trouver $n$, cela donne que le reste est contrôlé par $\\delta$ dès que $n \\leq \\log_{|x|}\\left( \\frac{6}{\\pi^2} \\delta \\right) - 1$ (si $x\\neq 0$, et par $n \\geq 0$ sinon)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import floor, log, pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "delta = 1e-5\n",
    "\n",
    "def f(x):\n",
    "    if x == 0: return 0\n",
    "    borne_sur_n = int(floor(log((6/pi**2 * delta), abs(x)) - 1))\n",
    "    somme_partielle = 0\n",
    "    for n in range(1, borne_sur_n + 1):\n",
    "        somme_partielle += x**n / n**2\n",
    "    return somme_partielle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pour x = -0.75 \tf(x) = -0.64276\n",
      "Pour x = -0.5 \tf(x) = -0.44841\n",
      "Pour x = 0.25 \tf(x) = 0.26765\n",
      "Pour x = 0 \tf(x) = 0\n",
      "Pour x = 0.25 \tf(x) = 0.26765\n",
      "Pour x = 0.5 \tf(x) = 0.58224\n",
      "Pour x = 0.75 \tf(x) = 0.97847\n"
     ]
    }
   ],
   "source": [
    "for x in [-0.75, -0.5, 0.25, 0, 0.25, 0.5, 0.75]:\n",
    "    print(\"Pour x =\", x, \"\\tf(x) =\", round(f(x), 5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- L'intégrale $g(t) = \\int_0^x \\frac{\\ln(1 - t)}{t} \\mathrm{d}t$ est bien défine sur $D = [-1, 1]$ puisque son intégrande existe, est continue et bien intégrable sur tout interval de la forme $]a, 0[$ ou $]0, b[$ pour $-1 < a < 0$ ou $0 < b < 1$. Le seul point qui peut déranger l'intégrabilité est en $0$, mais $\\ln(1-t) \\sim t$ quand $t\\to0$ donc l'intégrande est $\\sim 1$ en $0^-$ et $0^+$ et donc est bien intégrable. De plus, comme \"intégrale de la borne supérieure\" d'une fonction continue, $g$ est dérivable sur l'intérieur de son domaine, *i.e.*, sur $]-1, 1[$.\n",
    "\n",
    "- Pour la calculer numériquement, on utilise **évidemment** le module `scipy.integrate` et sa fonction `integrale, erreur = quad(f, a, b)`, qui donne une approximation de la valeur d'une intégrale en dimension 1 et une *borne* sur son erreur :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import integrate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "def g(x):\n",
    "    def h(t):\n",
    "        return log(1 - t) / t\n",
    "    integrale, erreur = integrate.quad(h, 0, x)\n",
    "    return integrale"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- On visualise les deux fonctions $f$ et $g$ sur le domaine $D$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fa9b59cf748>]"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fa9b5a17d30>]"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fa9b59cfe10>"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Représentation de $f(x)$ et $g(x)$')"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "domaine = np.linspace(-0.99, 0.99, 1000)\n",
    "\n",
    "valeurs_f = [f(x) for x in domaine]\n",
    "valeurs_g = [g(x) for x in domaine]\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(domaine, valeurs_f, label=\"$f(x)$\")\n",
    "plt.plot(domaine, valeurs_g, label=\"$g(x)$\")\n",
    "plt.legend()\n",
    "plt.grid()\n",
    "plt.title(\"Représentation de $f(x)$ et $g(x)$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- On conjecture que $g(x) = - f(x)$.\n",
    "\n",
    "La suite des questions est à faire au brouillon et sans Python :\n",
    "\n",
    "- On trouve que $f'(x) = \\sum_{n\\geq 1} \\frac{n x^{n-1}}{n^2} = \\frac{1}{x} \\sum_{n\\geq 1} \\frac{x^n}{n}$ si $x\\neq0$. Or on sait que $\\log(1 + x) = \\sum_{n\\geq 1} \\frac{x^n}{n}$ et donc cela montre bien que $g(x) = \\int_0^x - f'(t) \\mathrm{d}t = f(0) - f(x) = f(x)$ comme observé.\n",
    "\n",
    "- On trouve que $g(1) = - f(1) = - \\frac{\\pi^2}{6}$.\n",
    "\n",
    "- Par ailleurs, un changement de variable $u=1-x$ donne $g(1-x) = \\int_x^1 \\frac{\\ln(u)}{1-u} \\mathrm{d} u$, et une intégration par partie avec $a(u) = \\ln(u)$ et $b'(u) = \\frac{1}{1-u}$ donne $g(1-x) = [\\ln(u)\\ln(1-u)]_x^1 + \\int_x^1 \\frac{\\ln(1-u)}{u} \\mathrm{d}u$ et donc on reconnaît que $$g(1-x) = \\ln(x)\\ln(1-x) + g(1) - g(x).$$\n",
    "\n",
    "- Je vous laisse la fin comme exercice !"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "# À voir aussi\n",
    "\n",
    "## [Les oraux](http://perso.crans.org/besson/infoMP/oraux/solutions/)   *(exercices de maths avec Python)*\n",
    "\n",
    "Se préparer aux oraux de [\"maths avec Python\" (maths 2)](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/#oMat2) du concours Centrale Supélec peut être utile.\n",
    "\n",
    "Après les écrits et la fin de l'année, pour ceux qui seront admissibles à Centrale-Supélec, ils vous restera <b>les oraux</b> (le concours Centrale-Supélec a un <a title=\"Quelques exemples d'exercices sur le site du concours Centrale-Supélec\" href=\"http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/#oMat2\">oral d'informatique</a>, et un peu d'algorithmique et de Python peuvent en théorie être demandés à chaque oral de maths et de SI).\n",
    "\n",
    "Je vous invite à lire [cette page avec attention](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/#oMat2), et à jeter un œil aux documents mis à disposition :\n",
    "\n",
    "## Fiches de révisions *pour les oraux*\n",
    "\n",
    "1. [Calcul matriciel](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/Python-matrices.pdf), avec [numpy](https://docs.scipy.org/doc/numpy/) et [numpy.linalg](http://docs.scipy.org/doc/numpy/reference/routines.linalg.html),\n",
    "2. [Réalisation de tracés](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/Python-plot.pdf), avec [matplotlib](http://matplotlib.org/users/beginner.html),\n",
    "3. [Analyse numérique](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/Python-AN.pdf), avec [numpy](https://docs.scipy.org/doc/numpy/) et [scipy](http://docs.scipy.org/doc/scipy/reference/tutorial/index.html). Voir par exemple [scipy.integrate](http://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html) avec les fonctions [scipy.integrate.quad](http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html) (intégrale numérique) et [scipy.integrate.odeint](http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.odeint.html) (résolution numérique d'une équation différentielle),\n",
    "4. [Polynômes](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/Python-polynomes.pdf) : avec [numpy.polynomials](https://docs.scipy.org/doc/numpy/reference/routines.polynomials.package.html), [ce tutoriel peut aider](https://docs.scipy.org/doc/numpy/reference/routines.polynomials.classes.html),\n",
    "5. [Probabilités](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/Python-random.pdf), avec [numpy](https://docs.scipy.org/doc/numpy/) et [random](https://docs.python.org/3/library/random.html).\n",
    "\n",
    "Pour réviser : voir [ce tutoriel Matplotlib (en anglais)](http://www.labri.fr/perso/nrougier/teaching/matplotlib/), [ce tutoriel Numpy (en anglais)](http://www.labri.fr/perso/nrougier/teaching/numpy/numpy.html).\n",
    "Ainsi que tous les [TP](http://perso.crans.org/besson/infoMP/TPs/solutions/), [TD](http://perso.crans.org/besson/infoMP/TDs/solutions/) et [DS](http://perso.crans.org/besson/infoMP/DSs/solutions/) en Python que j'ai donné et corrigé au Lycée Lakanal (Sceaux, 92) en 2015-2016 !\n",
    "\n",
    "## Quelques exemples de sujets *d'oraux* corrigés\n",
    "> Ces 5 sujets sont corrigés, et nous les avons tous traité en classe durant les deux TP de révisions pour les oraux (10 et 11 juin).\n",
    "\n",
    "- PC : [sujet #1](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/PC-Mat2-2015-27.pdf) ([correction PC #1](http://perso.crans.org/besson/infoMP/oraux/solutions/PC_Mat2_2015_27.html)), [sujet #2](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/PC-Mat2-2015-28.pdf) ([correction PC #2](http://perso.crans.org/besson/infoMP/oraux/solutions/PC_Mat2_2015_28.html)).\n",
    "- PSI : [sujet #1](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/PSI-Mat2-2015-24.pdf) ([correction PSI #1](http://perso.crans.org/besson/infoMP/oraux/solutions/PSI_Mat2_2015_24.html)), [sujet #2](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/PSI-Mat2-2015-25.pdf) ([correction PSI #2](http://perso.crans.org/besson/infoMP/oraux/solutions/PSI_Mat2_2015_25.html)), [sujet #3](http://www.concours-centrale-supelec.fr/CentraleSupelec/MultiY/C2015/PSI-Mat2-2015-26.pdf) ([correction PSI #3](http://perso.crans.org/besson/infoMP/oraux/solutions/PSI_Mat2_2015_26.html)).\n",
    "- MP : pas de sujet mis à disposition, mais le programme est le même que pour les PC et PSI (pour cette épreuve)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## D'autres notebooks ?\n",
    "\n",
    "> Ce document est distribué [sous licence libre (MIT)](https://lbesson.mit-license.org/), comme [les autres notebooks](https://GitHub.com/Naereen/notebooks/) que j'ai écrit depuis 2015."
   ]
  }
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     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}