Limites de suites.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Limites de suites"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Recherche de seuils"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.1. Suite croissante de limite $+\\infty$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Prenons le cas d'une suite $(u_n)$ qui tend vers $+\\infty$.\n",
    "\n",
    "Quel que soit le réel A, il existe un rang $n_0$ tel que pour tout $n\\geq n_0$, $u_n\\geq A$.\n",
    "\n",
    "L'objectif est de déterminer un tel rang $n_0$ dans le cas d'une suite croissante."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
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    {
     "name": "stdout",
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     "text": [
      "Définissez la suite (u_n) grâce à la fonction précédente.\n",
      "Cel doit être une suite croissante qui tend vers +infini\n",
      "Quel seuil souhaitez-vous atteindre ?\n",
      "1000\n",
      "Les termes u_n dépasse le seuil donné dès que n >=  14 .\n"
     ]
    }
   ],
   "source": [
    "# Définition de la suite (u_n) à modifier si besoin\n",
    "\n",
    "def u(n):\n",
    "    if n == 0:\n",
    "        return 2\n",
    "    else:\n",
    "        return 1.5*u(n-1)+1\n",
    "\n",
    "# Détermination du premier rang pour lequel u(n) dépasse un seuil A (cas d'une suite croissante de limite +infini)\n",
    "\n",
    "def seuil1(A):\n",
    "    \"\"\"Détermination du premier rang pour lequel u(n) \n",
    "    dépasse le seuil A (cas d'une suite croissante de limite +infini)\"\"\"\n",
    "    n = 0\n",
    "    while u(n) < A:\n",
    "        n = n + 1\n",
    "    return n\n",
    "\n",
    "# Programme\n",
    "\n",
    "print(\"\"\"Définissez la suite (u_n) grâce à la fonction précédente.\n",
    "Ce doit être une suite croissante qui tend vers +infini\"\"\")\n",
    "A = float(input(\"Quel seuil souhaitez-vous atteindre ?\\n\"))\n",
    "print(\"Les termes u_n dépasse le seuil donné dès que n >= \", seuil1(A),\".\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.2. Suite décroissante de limite $-\\infty$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Prenons le cas d'une suite $(u_n)$ qui tend vers $-\\infty$.\n",
    "\n",
    "Quel que soit le réel A, il existe un rang $n_0$ tel que pour tout $n\\geq n_0$, $u_n\\leq A$.\n",
    "\n",
    "L'objectif est de déterminer un tel rang $n_0$ dans le cas d'une suite décroissante.\n",
    "\n",
    "**Travail à faire**\n",
    "\n",
    "Écrire une fonction `seuil2` déterminant le rang $n_0$ lié au seuil A.\n",
    "\n",
    "Pour cela, remplacez `pass` par un code qui convient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Définissez la suite (u_n) grâce à la fonction précédente.\n",
      "Ce doit être une suite décroissante qui tend vers -infini\n",
      "Quel seuil souhaitez-vous atteindre ?\n",
      "-1000\n",
      "Les termes u_n passe sous le seuil donné dès que n >=  10 .\n"
     ]
    }
   ],
   "source": [
    "# Définition de la suite (u_n) à modifier si besoin\n",
    "\n",
    "def u(n):\n",
    "    if n == 0:\n",
    "        return -2\n",
    "    else:\n",
    "        return 2*u(n-1)+1\n",
    "\n",
    "# Détermination du premier rang pour lequel u(n) dépasse un seuil A (cas d'une suite croissante de limite +infini)\n",
    "\n",
    "def seuil2(A):\n",
    "    \"\"\"Détermination du premier rang pour lequel u(n) \n",
    "    dépasse le seuil A (cas d'une suite croissante de limite +infini)\"\"\"\n",
    "    pass\n",
    "    return n\n",
    "\n",
    "# Programme\n",
    "\n",
    "print(\"\"\"Définissez la suite (u_n) grâce à la fonction précédente.\n",
    "Ce doit être une suite décroissante qui tend vers -infini\"\"\")\n",
    "A = float(input(\"Quel seuil souhaitez-vous atteindre ?\\n\"))\n",
    "print(\"Les termes u_n passe sous le seuil donné dès que n >= \", seuil2(A),\".\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Recherche de valeurs approchées"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.1. Valeur approchée de $\\pi$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Remarque :**\n",
    "\n",
    "Le nombre $\\pi$ facinnant les êtres humains depuis des siècles, il existe de nombreuses suites convergeant vers $\\pi$.\n",
    "\n",
    "Si vous souhaitez en savoir plus sur l'univers de $\\pi$, suivez le [lien](http://pi314.net/fr/index.php \"L'univers de pi\").\n",
    "\n",
    "Nous allons ici utiliser la méthode d'Archimède."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Explication de la méthode**\n",
    "\n",
    "La méthode d'Archimède permet d'obtenir une approximation du nombre $\\pi$. Pour cela on calcule les périmètres de polygones réguliers inscrits et circonscrits à un cercle de rayon $\\frac{1}{2}$. \n",
    "\n",
    "Plus le nombre de côtés du polygone sera important, plus on se rapprochera du périmètre du cercle, à savoir $\\pi$. \n",
    "\n",
    "![5, 6 et 8 côtés](https://media.eduscol.education.fr/ftp_eduscol/2019/Ressources/Mathematiques/img/archimede.png)\n",
    "\n",
    "Pour tout entier naturel $n\\geq 4$, on pose $a_n$ le périmètre d'un polygone régulier ayant $n$ côtés et inscrit dans le cercle de rayon $\\frac{1}{2}$ et $b_n$ le périmètre d'un polygone régulier ayant $n$ côtés et circonscrit au cercle de rayon $\\frac{1}{2}$. \n",
    "\n",
    "On vérifie que : $a_n=n\\sin\\left(\\dfrac{\\pi}{n}\\right)$ et $b_n=n\\tan\\left(\\dfrac{\\pi}{n}\\right)$.\n",
    "\n",
    "*Petit souci :* on cherche une approximation de $\\pi$. Donc utiliser $\\pi$ dans la méthode reste étrange.\n",
    "\n",
    "Pour tout entier naturel $n\\geq 2$, on pose $\\alpha_n=a_{2^n}$ et $\\beta_n=b_{2^n}$.\n",
    "\n",
    "On a également $\\displaystyle \\lim_{n\\to +\\infty} \\alpha_n=\\lim_{n\\to +\\infty} \\beta_n=\\pi$\n",
    "\n",
    "\n",
    "On peut démontrer que : $b_{2n}=\\dfrac{2a_nb_n}{a_n+b_n}$ et $a_{2n}=\\sqrt{a_nb_{2n}}$.\n",
    "\n",
    "On en déduit que $\\alpha_n=\\sqrt{\\alpha_{n-1}\\beta_n}$ et $\\beta_n=\\dfrac{2\\alpha_{n-1}\\beta_{n-1}}{\\alpha_{n-1}+\\beta_{n-1}}$.\n",
    "\n",
    "**Travail à faire**\n",
    "\n",
    "Écrire une fonction donnant un encadrement de $\\pi$ en utilisant les suites $(\\alpha_n)$ et $(\\beta_n)$.\n",
    "\n",
    "Pour cela, remplacez `pass` par un code qui convient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import sqrt\n",
    "\n",
    "def archimede(n):\n",
    "    a = 2*sqrt(2)\n",
    "    b = 4\n",
    "    for i in range(2,n):\n",
    "        pass\n",
    "    return a,b"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "**Une petite animation**"
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       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        animcb0f626d4b454298bfc9f1015723a652 = new Animation(frames, img_id, slider_id, 750.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# nombre d'itérations (on initialise au cas du carré)\n",
    "N = 4\n",
    "\n",
    "fig, (ax1, ax2) = plt.subplots(1, 2,figsize=(12, 6))\n",
    "l= .7\n",
    "ax1.set_xlim(( -l, l))\n",
    "ax1.set_ylim((-l, l))\n",
    "ax2.set_xlim(( 4, 2**(N+1)))\n",
    "ax2.set_ylim((2.75, 4.25))\n",
    "approximationsInf = []\n",
    "approximationsSup = []\n",
    "\n",
    "cercle = Circle((0, 0), .5, facecolor='none',\n",
    "                edgecolor=(0, 0, 0), linewidth=2, alpha=0.75)\n",
    "courbeInf, = ax2.plot([],[],'-o',color=\"#1e7fcb\")\n",
    "courbeSup, = ax2.plot([],[],'-o',color='orange')\n",
    "abscisse = []\n",
    "\n",
    "def init():\n",
    "    return []\n",
    "\n",
    "def animate(i):\n",
    "    abscisse.append(2**(i+2))\n",
    "    inf,sup = archimede(i+2)\n",
    "    approximationsInf.append(inf)\n",
    "    approximationsSup.append(sup)\n",
    "    ax1.clear()\n",
    "    ax1.set_xlim(( -l, l))\n",
    "    ax1.set_ylim((-l, l))\n",
    "    ax1.add_patch(cercle)\n",
    "    long = 0.5/cos(pi/(4*2**i))\n",
    "    PI = RegularPolygon(numVertices = 4*2**i,xy=(0, 0), radius=.5, orientation=0.79,edgecolor=\"#1e7fcb\", facecolor='none',\n",
    "                linewidth=2, alpha=0.5)\n",
    "    PS = RegularPolygon((0, 0), 4*2**i, radius=long, orientation=.79, facecolor='none',\n",
    "                edgecolor='orange', linewidth=2, alpha=0.5)\n",
    "    ax1.add_patch(PI)\n",
    "    ax1.add_patch(PS)\n",
    "    ax1.set_title('{} côtés'.format(4*2**i),color=\"#1e7fcb\",fontsize=14)\n",
    "    courbeInf.set_data(abscisse,approximationsInf)\n",
    "    courbeSup.set_data(abscisse,approximationsSup)\n",
    "    return PI,\n",
    "\n",
    "ax2.plot([4,2**(N+1)],[pi,pi],'--',color='green')\n",
    "ax2.legend(['Polygones intérieurs','Polygones extérieurs','$\\pi$'])\n",
    "\n",
    "plt.close ()\n",
    "ani = matplotlib.animation.FuncAnimation(fig, animate,init_func=init,  frames=N,blit=False,interval=750)\n",
    "# l'un ou l'autre\n",
    "HTML(ani.to_jshtml())\n",
    "#HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2. Valeur approchée de e"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Travail à faire**\n",
    "\n",
    "Programmer les algorithmes de l'exercice 65 p 67 du livre.\n",
    "\n",
    "Pour cela, remplacez `pass` par le code qui convient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Fonction définissant factorielle n\n",
    "\n",
    "pass\n",
    "\n",
    "# Fonction correspondant à la dernière question de l'exercice \n",
    "\n",
    "from math import e\n",
    "\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Remarque :**\n",
    "\n",
    "Python travaille avec 15 chiffres significatifs.\n",
    "\n",
    "Si vous souhaitez augmenter le nombre de chiffres significatifs utilisés par Python, \n",
    "suivez le [lien](http://villemin.gerard.free.fr/aInforma/PYTHON/PlusChif.htm \"Plus de chiffres significatifs\")."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.3. Valeur approchée de $\\sqrt{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On va utiliser la **suite de Héron** pour déterminer une approximation de $\\sqrt{2}$.\n",
    "\n",
    "Il s'agit de la suite $(u_n)$ définie sur $\\mathbb{N}$ par :\n",
    "\n",
    "$\\begin{cases}u_0=2\\\\u_{n+1}=\\dfrac{1}{2}\\left(u_n+\\dfrac{2}{u_n}\\right)\\text{ pour tout }n\\in\\mathbb{N}\\end{cases}$.\n",
    "\n",
    "Cette suite converge vers $\\sqrt{2}$.\n",
    "\n",
    "**Travail à faire**\n",
    "\n",
    "Écrire une fonction permettant de donner une approximation de $\\sqrt{2}$ à $10^{-p}$, $p$ entier choisi par l'utilisateur, et de donner l'indice de la suite.\n",
    "\n",
    "Pour cela, remplacez `pass` par le code qui convient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import sqrt\n",
    "\n",
    "def heron(p):\n",
    "    u = 2\n",
    "    pass\n",
    "    return(u,n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.4. Valeur approchée de $\\dfrac{1+\\sqrt{5}}{2}$ (nombre d'or)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Travail à faire**\n",
    "\n",
    "Programmer les algorithmes de l'exercice 74 p 69 du livre.\n",
    "\n",
    "Pour cela, remplacez `pass` par le code qui convient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Programme correspondant à la suite de la partie B\n",
    "\n",
    "from math import sqrt\n",
    "\n",
    "pass\n",
    "\n",
    "\n",
    "# Programme correspondant à la suite de la partoe C\n",
    "\n",
    "pass"
   ]
  }
 ],
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