{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "c47b6799",
   "metadata": {},
   "source": [
    "# Feuille de calcul symbolique de développements limités "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a7bc68b7",
   "metadata": {},
   "source": [
    "Importation des packages Python"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b41024c8",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "from sympy.interactive import printing "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d3468afe",
   "metadata": {},
   "source": [
    "x est désormais un symbole (qui sera la variable dans nos DL)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "9b93e6b6",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols('x')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "54ed43ba",
   "metadata": {},
   "source": [
    "La fonction series permet de faire des DL.\n",
    "Par défaut, les DL sont calculés en 0 à l'ordre 5.\n",
    "Attention : Python utilise un \"grand O\" et pas un \"petit o\". "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "58cab3ff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x - \\frac{x^{2}}{2} + \\frac{x^{3}}{3} - \\frac{x^{4}}{4} + \\frac{x^{5}}{5} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(ln(1+x),x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1553eae7",
   "metadata": {},
   "source": [
    "On écrira que $\\ln(1+x)= x-\\dfrac{x^2}{2}+ \\dfrac{x^3}{3}-\\dfrac{x^4}{4}+\\dfrac{x^5}{5}+o(x^5)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "affa29e4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "1 - x**2/2 + x**4/24 + O(x**6)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(cos(x),x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "000849cc",
   "metadata": {},
   "source": [
    "On écrira que $\\cos(x)= 1-\\dfrac{x^2}{2}+ \\dfrac{x^4}{24}+o(x^5)$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92b1e9e5",
   "metadata": {},
   "source": [
    "Pour faire un DL au point c à un ordre supérieur à 6, exécuter le code ci-dessous."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "67cf9660",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{2} - \\frac{\\left(x - 1\\right)^{2}}{8} + \\frac{\\left(x - 1\\right)^{3}}{16} - \\frac{5 \\left(x - 1\\right)^{4}}{128} + \\frac{7 \\left(x - 1\\right)^{5}}{256} - \\frac{21 \\left(x - 1\\right)^{6}}{1024} + \\frac{33 \\left(x - 1\\right)^{7}}{2048} + \\frac{x}{2} + O\\left(\\left(x - 1\\right)^{8}; x\\rightarrow 1\\right)$"
      ],
      "text/plain": [
       "1/2 - (x - 1)**2/8 + (x - 1)**3/16 - 5*(x - 1)**4/128 + 7*(x - 1)**5/256 - 21*(x - 1)**6/1024 + 33*(x - 1)**7/2048 + x/2 + O((x - 1)**8, (x, 1))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c=1\n",
    "n=7\n",
    "series(sqrt(x),x,c,n+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5b042275",
   "metadata": {},
   "source": [
    "Pensez à remplacer le $O((x-1)^8)$ par un $o((x-1)^7)$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c85bb61a",
   "metadata": {},
   "source": [
    "## Corrections de la feuille de TD 9"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b338d531",
   "metadata": {},
   "source": [
    "Exercice 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "d8bc9dd3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1 + 2 x - \\frac{3 x^{2}}{2} + \\frac{2 x^{3}}{3} - \\frac{11 x^{4}}{24} + O\\left(x^{5}\\right)$"
      ],
      "text/plain": [
       "1 + 2*x - 3*x**2/2 + 2*x**3/3 - 11*x**4/24 + O(x**5)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 1.\n",
    "series(cos(x)+2*ln(1+x),x,0,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d913fa9f",
   "metadata": {},
   "source": [
    "On obtient $\\cos(x)+2\\ln(1+x)=1+2x-\\dfrac{3x^2}{2}+ \\dfrac{2x^3}{3}-\\dfrac{11x^4}{24}+o(x^4)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "996cf4c0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1 - \\frac{x}{2} - \\frac{x^{2}}{8} + \\frac{11 x^{3}}{48} - \\frac{5 x^{4}}{128} + \\frac{73 x^{5}}{3840} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "1 - x/2 - x**2/8 + 11*x**3/48 - 5*x**4/128 + 73*x**5/3840 + O(x**6)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#2.\n",
    "series(sqrt(1+x)-sin(x),x,0,6)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8ac973a1",
   "metadata": {},
   "source": [
    "On obtient $\\sqrt{1+x}-\\sin(x)=1-\\dfrac{x}{2}-\\dfrac{x^2}{8}+ \\dfrac{11x^3}{48}-\\dfrac{5x^4}{128}+ \\dfrac{73x^5}{3840}+o(x^5)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "34295d0a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 x + \\frac{2 x^{3}}{3} + O\\left(x^{5}\\right)$"
      ],
      "text/plain": [
       "2*x + 2*x**3/3 + O(x**5)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#3.\n",
    "series(x+x**3+atan(x),x,0,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "428c0684",
   "metadata": {},
   "source": [
    "On obtient $x+x^3+\\arctan(x)=2x+ \\dfrac{2x^3}{3}+o(x^4)$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c7f3f1a",
   "metadata": {},
   "source": [
    "Exercice 6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "532db4ef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{2} - \\frac{\\left(x - 1\\right)^{2}}{8} + \\frac{\\left(x - 1\\right)^{3}}{16} + \\frac{x}{2} + O\\left(\\left(x - 1\\right)^{4}; x\\rightarrow 1\\right)$"
      ],
      "text/plain": [
       "1/2 - (x - 1)**2/8 + (x - 1)**3/16 + x/2 + O((x - 1)**4, (x, 1))"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#1.\n",
    "series(sqrt(x),x,1,4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71f4b942",
   "metadata": {},
   "source": [
    "On obtient $\\sqrt{x} = \\dfrac{1}{2}- \\dfrac{(x-1)^2}{8}+\\dfrac{(x-1)^3}{16}+\\dfrac{x}{2}+o((x-1)^3)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "42ce6f43",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -1 - \\frac{\\left(x - 1\\right)^{2}}{2} + \\frac{\\left(x - 1\\right)^{3}}{2} + x + O\\left(\\left(x - 1\\right)^{4}; x\\rightarrow 1\\right)$"
      ],
      "text/plain": [
       "-1 - (x - 1)**2/2 + (x - 1)**3/2 + x + O((x - 1)**4, (x, 1))"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#2.\n",
    "series(asin(ln(x)),x,1,4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c832729a",
   "metadata": {},
   "source": [
    "On obtient $\\arcsin(\\ln(x)) = -1-\\dfrac{(x-1)^2}{2}+\\dfrac{(x-1)^3}{2}+x+o((x-1)^3)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "6e15c5d6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e^{\\frac{\\sqrt{3}}{2}} + \\frac{\\left(x - \\frac{\\pi}{3}\\right) e^{\\frac{\\sqrt{3}}{2}}}{2} + \\left(x - \\frac{\\pi}{3}\\right)^{2} \\left(- \\frac{\\sqrt{3} e^{\\frac{\\sqrt{3}}{2}}}{4} + \\frac{e^{\\frac{\\sqrt{3}}{2}}}{8}\\right) + \\left(x - \\frac{\\pi}{3}\\right)^{3} \\left(- \\frac{\\sqrt{3} e^{\\frac{\\sqrt{3}}{2}}}{8} - \\frac{e^{\\frac{\\sqrt{3}}{2}}}{16}\\right) + O\\left(\\left(x - \\frac{\\pi}{3}\\right)^{4}; x\\rightarrow \\frac{\\pi}{3}\\right)$"
      ],
      "text/plain": [
       "exp(sqrt(3)/2) + (x - pi/3)*exp(sqrt(3)/2)/2 + (x - pi/3)**2*(-sqrt(3)*exp(sqrt(3)/2)/4 + exp(sqrt(3)/2)/8) + (x - pi/3)**3*(-sqrt(3)*exp(sqrt(3)/2)/8 - exp(sqrt(3)/2)/16) + O((x - pi/3)**4, (x, pi/3))"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#3.\n",
    "series(exp(sin(x)),x,pi/3,4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "550b6927",
   "metadata": {},
   "source": [
    "Exercice 7 \n",
    "\n",
    "Remarque : penser à remplacer les $O(x^{n+1})$ par des $o(x^n)$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "d8af9e4e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1 + x - \\frac{x^{3}}{3} - \\frac{x^{4}}{6} + O\\left(x^{5}\\right)$"
      ],
      "text/plain": [
       "1 + x - x**3/3 - x**4/6 + O(x**5)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#1.\n",
    "series(cos(x)*exp(x),x,0,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "3c3fea13",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{6} + O\\left(x^{7}\\right)$"
      ],
      "text/plain": [
       "x**6 + O(x**7)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#2.\n",
    "series((sin(x))**6,x,0,7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "14d7554c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} - x^{3} + \\frac{11 x^{4}}{12} + O\\left(x^{5}\\right)$"
      ],
      "text/plain": [
       "x**2 - x**3 + 11*x**4/12 + O(x**5)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#3.\n",
    "series((ln(1+x))**2,x,0,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "03d38bbf",
   "metadata": {},
   "source": [
    "Exercice 8 \n",
    "\n",
    "Remarque : penser à remplacer les $O(x^{n+1})$ par des $o(x^n)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "5ca26140",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1 + \\frac{x^{2}}{2} + \\frac{5 x^{4}}{24} + O\\left(x^{5}\\right)$"
      ],
      "text/plain": [
       "1 + x**2/2 + 5*x**4/24 + O(x**5)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#1.\n",
    "series(1/cos(x),x,0,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "974797ce",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + \\frac{x^{3}}{3} + \\frac{2 x^{5}}{15} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "x + x**3/3 + 2*x**5/15 + O(x**6)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#2.\n",
    "series(tan(x),x,0,6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "5a3f6414",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\log{\\left(4 \\right)} + \\frac{x}{2} + \\frac{3 x^{2}}{8} - \\frac{x^{3}}{8} - \\frac{x^{4}}{64} + \\frac{x^{5}}{32} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "log(4) + x/2 + 3*x**2/8 - x**3/8 - x**4/64 + x**5/32 + O(x**6)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#3.\n",
    "series(ln(3*exp(x)+exp(-x)),x,0,6)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8498dc2f",
   "metadata": {},
   "source": [
    "Exercice 11 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "7a501b81",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{1}{2} + \\frac{23 x^{2}}{24} - \\frac{719 x^{4}}{720} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "-1/2 + 23*x**2/24 - 719*x**4/720 + O(x**6)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 1.\n",
    "series((1/(1+x**2)-cos(x))/x**2,x,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "382e2535",
   "metadata": {},
   "source": [
    "On en déduit que $\\lim_{x\\to 0}f(x) = -\\dfrac{1}{2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "233e48f7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 - \\frac{11 x^{2}}{10} + \\frac{3359 x^{4}}{4200} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "2 - 11*x**2/10 + 3359*x**4/4200 + O(x**6)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 2.\n",
    "series((atan(x)-x)/(sin(x)-x),x,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fca5e0a6",
   "metadata": {},
   "source": [
    "On en déduit que $\\lim_{x\\to 0}f(x) = 2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "41511bf5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2}{3} + \\frac{4 x^{4}}{189} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "2/3 + 4*x**4/189 + O(x**6)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 3.\n",
    "series(1/(sin(x))**2 - 1/(sinh(x))**2,x,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "59613413",
   "metadata": {},
   "source": [
    "On en déduit que $\\lim_{x\\to 0}f(x) = 2/3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "383e2d72",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\log{\\left(b \\right)}}{2} + \\frac{\\log{\\left(a \\right)}}{2} + x \\left(- \\frac{\\left(\\frac{\\log{\\left(a \\right)}}{2} + \\frac{\\log{\\left(b \\right)}}{2}\\right)^{2}}{2} + \\frac{\\log{\\left(a \\right)}^{2}}{4} + \\frac{\\log{\\left(b \\right)}^{2}}{4}\\right) + O\\left(x^{2}\\right)$"
      ],
      "text/plain": [
       "log(b)/2 + log(a)/2 + x*(-(log(a)/2 + log(b)/2)**2/2 + log(a)**2/4 + log(b)**2/4) + O(x**2)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#4.\n",
    "a = symbols('a')\n",
    "b = symbols('b')\n",
    "# La technique est de faire un DL de ln(f(x)).\n",
    "# on tronque le DL pour n'avoir que des termes d'ordre 0 et 1.\n",
    "series(ln((a**x +b**x)/2)*(1/x),x,0,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d88a94dd",
   "metadata": {},
   "source": [
    "En repassant à l'exponentielle, on en déduit que $\\lim_{x\\to 0} f(x) = \\sqrt{ab}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1e263d0c",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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