{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "f7103e0f-bcae-4297-b550-31fe911dffa1",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "88baf7ed-203d-4960-b314-413828ce29b1",
   "metadata": {},
   "outputs": [],
   "source": [
    "def rectangle( x ): \n",
    "    return np.abs(x) < 1/4\n",
    "\n",
    "def sum_to(upper, x):\n",
    "    return 0 + sum( 0*np.cos(2*np.pi*n*x) + 0*np.sin(2*np.pi*n*x)  for n in range(1, upper + 1) ) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "32ec6b94-fbb5-4994-8d96-e8b3d1900299",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f0ea0f63340>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "fig, ax = plt.subplots(1,1)\n",
    "x = np.linspace(-1, 1, 1000)\n",
    "for upper in [5, 10, 100]:\n",
    "    ax.plot( x, rectangle(x) )\n",
    "    ax.plot( x, sum_to(upper, x), label = f'$n = {upper}$' )\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "04a3eed4-827a-4df9-8ce1-5e80ff33c749",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.9.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}