{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# レポート課題の解答例\n",
    "\n",
    "2021年5月9日　池 浩一郎\n",
    "\n",
    "## 級数と調和数列\n",
    "\n",
    "実数$p$ ($p>0$)に対して、数列$\\{a(k)\\}$を以下のように定義します。\n",
    "\n",
    "$$\n",
    "a(k)=\\frac{1}{k^p}, \\quad k=1,2,3,\\cdots\n",
    "$$\n",
    "\n",
    "数列$\\{a(k)\\}$による級数$\\sum_{n=1}^\\infty a(n) $を定義します。級数の第$n$部分和$S(n)$は以下の式で計算できます。\n",
    "\n",
    "$$S(n) = \\sum_{k=1}^n a(k) = 1+\\frac{1}{2^p}+\\frac{1}{3^p}+\\frac{1}{4^p} + \\cdots + \\frac{1}{n^p}$$\n",
    "\n",
    "特に$p=1$の時、$\\{a(k)\\}$は調和数列になります。\n",
    "\n",
    "\n",
    "### 演習３\n",
    "\n",
    "- 3.1) 数列$\\{a(k)\\}$を作成してください。ただし、$k$を50までとします。\n",
    "- 3.2) $p=0.1, 0.5,1,2,4$に対して、数列$\\{S(n)\\}$ ($n=1,\\cdots, 50$)のグラフを描いて、数列が収束するかどうかを考察してください。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "max_N=50\n",
    "n_list=range(1,max_N+1)\n",
    "#for p in [0.1,0.5,1,2,4]:\n",
    "for p in [1]:\n",
    "    a_list=[]\n",
    "    for n in n_list:\n",
    "        a_list.append(1/pow(n,p))\n",
    "\n",
    "    S_list=[]\n",
    "    sum=0\n",
    "    for a in a_list:\n",
    "        sum = sum + a\n",
    "        S_list.append(sum)\n",
    "    \n",
    "    plt.plot(S_list,'.-') #x軸の数列を省略することが可能です。この場合、Pythonが自動的に設定してくれます。\n",
    "    plt.title(\"p=%5.2f\"%p)\n",
    "    plt.grid()\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "次のコードでは、全てのグラフを一つの図に描いてみます。labelとlegendを組み合わせて使用することで、各曲線にラベルを付けることができます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "max_N=50\n",
    "n_list=range(1,max_N+1)\n",
    "\n",
    "for p in [0.1,0.5,1,2,4]:\n",
    "    a_list=[]\n",
    "    for n in n_list:\n",
    "        a_list.append(1/pow(n,p))\n",
    "\n",
    "    S_list=[]\n",
    "    sum=0\n",
    "    for a in a_list:\n",
    "        sum = sum + a\n",
    "        S_list.append(sum)\n",
    "    \n",
    "    plt.plot(S_list,'-',label=\"p=%5.2f\"%p)\n",
    "    \n",
    "plt.title(\"Graph of sequence S(n) for different p\")\n",
    "plt.legend()\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 調和級数の発散について\n",
    "\n",
    "$p=1$のとき、$\\{S_n\\}$が発散することを証明してみます。\n",
    "\n",
    "$$\n",
    "\\sum_{n=1}^{\\infty}\\frac{1}{n} = 1 + \\frac{1}{2}+ \\frac{1}{3} + \\cdots =\n",
    "1+\\frac{1}{2}+\\left(\\frac{1}{3}+\\frac{1}{4}\\right) + \\left(\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{7}+\\frac{1}{8}\\right) + \\cdots\n",
    "$$\n",
    "\n",
    "ここで、以下の不等式に注意してください。\n",
    "$$\n",
    "\\frac{1}{3}+\\frac{1}{4} \\ge \\frac{1}{4}+\\frac{1}{4} = \\frac{1}{2}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{7}+\\frac{1}{8} \\ge \\frac{1}{8}+\\frac{1}{8}+\\frac{1}{8}+\\frac{1}{8} = \\frac{1}{2}\n",
    "$$\n",
    "\n",
    "よって、\n",
    "\n",
    "$$\n",
    "\\lim_{n\\to \\infty}S(n) = \\sum_{n=1}^{\\infty} \\frac{1}{n} \\ge 1 + \\frac{1}{2}+  \\frac{1}{2} +  \\frac{1}{2} + \\cdots = \\infty\n",
    "$$\n",
    "\n",
    "\n",
    "## 一般の$p$に対する級数の収束・発散について\n",
    "\n",
    "$0<p\\le 1$のとき、$\\{S_n\\}$は発散します。また、$p>1$のとき、$\\{S_n\\}$は収束します。この結果について、以下の関数の積分を利用して証明することができます。その詳細はここでは省略します。\n",
    "\n",
    "- $p>1$ のとき\n",
    "$$\n",
    "\\sum_{k=2}^\\infty \\frac{1}{k^p} = \\frac{1}{2^p}+ \\frac{1}{3^p} + \\cdots  \\le  \\int_{1}^\\infty \\frac{1}{x^p} dx = \\frac{1}{p-1} <\\infty\n",
    "$$"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}
