3.Real Analysis.ipynb

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    "EE 502 P: Analytical Methods\n",
    "# 3: Real Analysis\n",
    "## Prof. Eric Klavins, Electrical and Computer Engineering\n",
    "25 September, 2019\n",
    "Copyright &copy; 2019, University of Washington\n",
    "\n",
    "![](https://mathwithbaddrawings.files.wordpress.com/2015/02/20150225072534_00001.jpg)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Outline\n",
    "\n",
    "Limits of Sequences\n",
    "- Definition\n",
    "- Examples: $n^2$, $1/n$\n",
    "- Sequence convergence\n",
    "- Examples: $1/n$, $1/(1+rn)$\n",
    "\n",
    "Limits of functions\n",
    "- Cluster points and relationship to sequences\n",
    "- $\\varepsilon-\\delta$ definition\n",
    "- Examples: $b$, $x^2$, $1/x$ as $x \\rightarrow c \\neq 0$\n",
    "- Sequence definition and divergence criteria\n",
    "- Examples: $1/x$ as $x \\rightarrow 0$\n",
    "- Rules for sum, difference, product, multiple, quotient\n",
    "- Examples: $1/x$, poly / poly, \n",
    "- Not covered: limits at infinity, one-sided limits\n",
    "\n",
    "Continuous Functions\n",
    "- Neighborhood and sequence definitions\n",
    "- Examples: $b$, $x$, $1/x$, $\\sin(1/x)$\n",
    "- Combinations of functions (sum, multiple, produce, quotient, composition, inverses)\n",
    "- $\\sin$, $\\cos$ (not proven to save time)\n",
    "- Intermediate value theorem (and restatements)\n",
    "- Examples: Root finding via bisection (see homework)\n",
    "\n",
    "Derivatives\n",
    "- Definition in terms of limit\n",
    "- Combinations: sum, multiple, product, quotient, infinite sums and products\n",
    "- Chain rule\n",
    "- Examples: 1/f, tan\n",
    "- Mean value theorem\n",
    "- Example: Finding square roots\n",
    "- Example: Showing $e^x \\geq 1+x$, $(1+x)^a \\geq 1 + ax$, etc\n",
    "- L'Hospital's rules and examples of various indeterminate forms (see homework)\n",
    "\n",
    "Taylor's Theorem\n",
    "- Statement, derivation, and short proof\n",
    "- Examples: cube roots, sin, and exponential\n",
    "\n",
    "Integrals\n",
    "- Will do a homework exercise on"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:19.635967Z",
     "start_time": "2019-10-09T22:14:19.630850Z"
    },
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "import sympy as sm\n",
    "sm.init_printing(use_latex='mathjax')\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.patches as pch\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-07T21:27:21.854337Z",
     "start_time": "2019-10-07T21:27:21.851668Z"
    }
   },
   "source": [
    "# Homework 1 Results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:21.725455Z",
     "start_time": "2019-10-09T22:14:21.479854Z"
    }
   },
   "outputs": [
    {
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26JpjSNKVZNNCnFlkYGZmVo689xGMrFnevIhAzMysNfK0CL4ELJT0C7JLSA8ETi80KjMz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\nnAjMzCrufwCNycayULfR9wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hw1 =[ 80,80,80,80,80,80,80,80,80,80,80,80,\n",
    "       80,79,78,78,78,78,76,75,75,75,75,74,\n",
    "       74,74,74,74,73,73,73,73,73,71,71,71,\n",
    "       70,70,70,69,68.5,68,67,67,67,66.5,66,\n",
    "       64,58,55,53,50.5]\n",
    "\n",
    "plt.hist(hw1,bins=range(45,81));\n",
    "plt.title(\"Homework 1 Scores\")\n",
    "plt.xlabel(\"Score\")\n",
    "plt.ylabel(\"Number of Students\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sequences"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Def:** A **sequence** of real numbers is a real valued function on the set $\\mathbb{N} = \\{ 0, 1, 2, 3, \\dots \\}$. Sometimes we include $0$ in $\\mathbb{N}$, sometimes we don't. It should be clear from context. \n",
    "\n",
    "If $X : \\mathbb{N} \\rightarrow \\mathbb{R}$, the sequence of values is referred to by\n",
    "\n",
    "$$\n",
    "X(0), X(1), X(2), \\dots\n",
    "$$\n",
    "\n",
    "Usually, we refer to the elements of the sequence by $x_n = X(n)$ and write\n",
    "\n",
    "$$\n",
    "x_0, x_1, x_2, \\dots\n",
    "$$\n",
    "\n",
    "For example, if $x_n = n^2$, then\n",
    "\n",
    "$$\n",
    "X = ( 0, 1, 4, 9, \\dots )\n",
    "$$\n",
    "\n",
    "In Python we represent sequences as functions that take a single integer $n$ and return the value at that $n$. So that we can manipulate them like sequences, we can wrap them in a `np.array`. So that we can do exact math, we might also use `sympy` to represent the elements of the sequence."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-10T02:14:30.652666Z",
     "start_time": "2019-10-10T02:14:30.635985Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "([1,\n",
       "  4,\n",
       "  9,\n",
       "  16,\n",
       "  25,\n",
       "  36,\n",
       "  49,\n",
       "  64,\n",
       "  81,\n",
       "  100,\n",
       "  121,\n",
       "  144,\n",
       "  169,\n",
       "  196,\n",
       "  225,\n",
       "  256,\n",
       "  289,\n",
       "  324,\n",
       "  361],\n",
       " array([  1,   4,   9,  16,  25,  36,  49,  64,  81, 100, 121, 144, 169,\n",
       "        196, 225, 256, 289, 324, 361]),\n",
       " array([1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256,\n",
       "        289, 324, 361], dtype=object))"
      ]
     },
     "execution_count": 180,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#\n",
    "# Example: The sequence (n**2 : for n \\in N)\n",
    "#\n",
    "\n",
    "def x(n):\n",
    "    return n**2\n",
    "\n",
    "# As a Python list\n",
    "X1 = [x(n) for n in range(1,20)]\n",
    "\n",
    "# As a numpy list\n",
    "X2 = np.array([x(n) for n in range(1,20)])\n",
    "\n",
    "# As a numpy list of infinite precesion sympy numbers\n",
    "X = np.array([x(sm.Integer(n)) for n in range(1,20)])\n",
    "X1, X2, X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-10T02:16:02.672249Z",
     "start_time": "2019-10-10T02:16:02.668058Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(sympy.core.numbers.One, numpy.ndarray)"
      ]
     },
     "execution_count": 185,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(X[0]), type(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-10T02:17:05.746211Z",
     "start_time": "2019-10-10T02:17:05.739524Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13,\n",
       "       1/14, 1/15, 1/16, 1/17, 1/18, 1/19], dtype=object)"
      ]
     },
     "execution_count": 187,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#\n",
    "# Example: The sequence (1/n : for n \\in N). This better shows why we might use sympy.\n",
    "#\n",
    "\n",
    "def y(n):\n",
    "    return 1/n\n",
    "\n",
    "Y = np.array([y(sm.Integer(n)) for n in range(1,20)])\n",
    "Y\n",
    "\n",
    "# Y = np.array([y(n) for n in range(1,20)])\n",
    "# Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Combining sequences\n",
    "\n",
    "If $(x_n)$ and $(y_n)$ represent sequences, the we can construct new sequences by adding, subtracting, multiplying, and dividing them elementwise to get\n",
    "\n",
    "$$\n",
    "(x_n + y_n)\n",
    "$$\n",
    "\n",
    "$$\n",
    "(x_n - y_n)\n",
    "$$\n",
    "\n",
    "$$\n",
    "(x_n y_n)\n",
    "$$\n",
    "\n",
    "$$\n",
    "(x_n / y_n)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:28.329576Z",
     "start_time": "2019-10-09T22:14:28.322226Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2 9/2 28/3 65/4 126/5 217/6 344/7 513/8 730/9 1001/10 1332/11 1729/12\n",
      " 2198/13 2745/14 3376/15 4097/16 4914/17 5833/18 6860/19] \n",
      "\n",
      "[0 7/2 26/3 63/4 124/5 215/6 342/7 511/8 728/9 999/10 1330/11 1727/12\n",
      " 2196/13 2743/14 3374/15 4095/16 4912/17 5831/18 6858/19] \n",
      "\n",
      "[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19] \n",
      "\n",
      "[1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 4096 4913\n",
      " 5832 6859] \n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(X+Y, \"\\n\")\n",
    "print(X-Y, \"\\n\")\n",
    "print(X*Y, \"\\n\")\n",
    "print(X/Y, \"\\n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:29.707622Z",
     "start_time": "2019-10-09T22:14:29.037688Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Sfl/SwaO2vyJdsvZqSe/LKz6zVtSx/Tdb6OZNTfb/qqTXdzYks+6o0gIWPfJb\nwH7AjZK2Srpf0mMky9a+D7g8Iv4y1wjNMqpj+29Won+npM8C3wE28PJa1K8B3kwybe6jXY3QrEOq\ntIBFL0TEc8CXgC9JeiXw68DuiPhxvpGZta6O7b/ZQjcfkXQQ8E7gXcAMkrXoHwGuiIi/636IZp1T\nlQUsekHSt4DzIuKfI+J54KkWXnsocC0wnWS8zqqIuDTtT75GsireFuCMiHhWkoBLgdOA54BzIuK+\nTv4+ZnVr/81K9LMiYmdEXBkR50TEwohYEhGfaJbcJR2aXsN7WNJDkj6Ubj8oXcP+sfT7gel2SfqS\npM1pKfCYzv2aZtaGrwJ3SPq4pClNj97bHuCjEXEkyUI550k6ElgB3BERc4A70ucAbwPmpF/LSG5y\nY2aT0KxE/11JVwGfi4g9Lb73SAO/T9KrgA2SbgfOIWngKyWtIGngH2fvBn48SQM/vsWfaTVVtwUs\neiEirpP0N8AngSFJa4BfNOz/0gSv3QZsSx//TNIjwACwGDg5PWwNcBdJ+18MXJveevYeSf2SZqTv\nY9aU+4B9NRtFP4+kxLZB0omtvHF6t6n70sc/IynrjzTwNelha4Al6eOXGnhE3AP0p7ekNJvQyAIW\nw7t2E7y8gMXajcN5h1YFu4GfAP+OZDW7xq9M0vvKzyO5I+X0hqT9NEn/Aknf0HgJYGu6zawp9wFj\na3YN/mfARyQdS1Kq20ryCV7J7sh0q9hJNvC9PsFLWkZSwmPmTN/Izuq5gEUvSDqF5Lr4d4BjI+Jf\n23iPXwH+GvhwRPxLcqk9EREhKdp4T/cBthf3AWNrutCNpAUkjfwq4DIaSnRZdLqBR8QqYBXA4OBg\ny52DVU8dF7DokU8BvxcR/7edF0vaj6Tt/1VEfDPd/MxI6T2t0I3cjXIYOLTh5Yek2/bhPsBGcx8w\ntmaD7K4HLgTeExH/NSKeiIgnR76avflEDTzd31YDN2tUxwUseiEi3jg6uadnz02lo+KvBh6JiC80\n7FrHyytjLgVuatj+3nSw7QnAT3393bJyHzC2ZtfgvxsRb4qIB1p9Yzdw65U6LmCRo/8v43HzgbOB\nBZI2pV+nASuBt6YL5rwlfQ5wC/A4sJnkrpV/0NmwrcrcB4yt2TX4qxqfS3prRNye8b1HGvgDkjal\n2/6EpEHfIOlc4EngjHTfLSRzYDeTzIP1EpiWSR0XsMiRmh8C6TTa8Y49ZYzjAzhvEnFZjbkPGJuS\ndpXxYOm+iCjM/PTBwcEYGhrKOwyznpK0ISIGc/rZh0TE1jx+9ljcB1jdtNL+s95Nziw3nt9aDC1W\n8Mw6wu2/fVlG0f8lyVKTAmb8BMAKAAAgAElEQVRKWj2yLyLe38XYzF6a3zoyBWZkfivgRt57nwGc\n4K1n3P4nJ8sZ/DUNj9/Ey4vUmHWd57ea1Zfb/+Q0TfAR8bcjjyX9rPG5Wbd5fmu+XMGzPLn9T06r\n1+Bf6EoUZuM4uL+P4TEac93nt/bQNQ2PXcGznnL7n5xm8+D3EhEndCsQs7F4fmu+IuJvR76An416\nbtZVbv+Tk/kMPr2t68EkN5/YEhEtLVlr1g7Pby0UV/Csp9z+J2fCBC/p1SSLT5wF7A/sAA4Apku6\nB/hyRNzZ9Sit1pbMG3CDLgBX8KzTskyBc/tvX7Mz+G8A1wInRsSuxh3pHebOlnR4RFzdrQCtujy/\ntTxcwbNO8xS47mu2VO1bJ9i3AdjQ8YisFty4i88VPOsmT4HrvlauwQ8AhzW+JiLu7kZQVn1u3KXg\nCp51jafAdV+mBC/pM8C7gYeBkV45ACd4a4sbd/G5gmfd5Clw3Zf1DH4JMDcinu9mMFYfbtzl4gqe\nddryhXP3ukwHngLXaVkT/OPAfoATvHWEG3d5uIJn3eApcN2XNcE/B2ySdAcNST4i/qgrUVnpNRsh\n78ZdKq7gWUuyzpDxFLjuyprg16VfZk1lHSHvxl0aruBZZp4hUxyZEnxEeP1py8wj5CvHFTzLzO2/\nOJqtZPctYBXwnYj4+ah9hwPnkCx6sXqMl1tNeYR85biCZ5m5/RdHszP4DwB/DHxR0k5eXuhiNrAZ\n+POIuKm7IVrZeIR8tbiCZ61w+y+OZneT2z8iPhYRRwDvAj5FkvBfB1zo5G5j8R2gqkHStyT9tqT9\nxth3uKQLJfme8LYXt//iaJbg75L0MUlTImJLRHwf2EZStr9kohdKWi1pu6QHG7YdJOl2SY+l3w9M\nt0vSlyRtlnS/pGMm+4tZfpbMG+Ci049ioL8PAQP9fVx0+lG+/lY+HwBOBP5B0r2SbpG0XtITwBXA\nhokuz7kPqJ61G4eZv3I9s1fczPyV61m7cXifY9z+i0MRMf7OpPGtBN4IfAg4iuQM/rPA5RPdcELS\nScC/AtdGxOvTbZ8FdkbESkkrgAMj4uOSTgP+EDgNOB64NCKObxb84OBgDA0NZftNrSN8g5j8SdoQ\nEYM9+DkzI+Kf08ezgBkkN5v5R+DYiPg/TV7vPqBCRo+Oh+TM3Mm7t1pp/xOewUfEsxHxQeAq4LvA\ncmB+RFzW7G5S6SpXO0dtXgyMXM9bQzK/dmT7tZG4B+iXNCPLL2C9M9LAh3ftJnh5+stYn+KtEtqu\n4IH7gKqZaHS8FdOECV5Sv6QrgPcBi0huPvFtSQva/HnTI2Jb+vhpYHr6eAB4quG4rek2KxA38No5\nFjiCZIrcAkkfAn4IfB84rs33nHQfIGmZpCFJQzt27GgzDGuVR8eXT7NR9PcBXwbOi4g9wG2Sjga+\nLOnJiDir3R8cESFp/OsD45C0DFgGMHPmzHZ/vLXBDbxeIuJZ4INpYv8u8CPghIjY2qH3b6sPiIhV\nJFUEBgcHW369tcej48un2SC7kyLic2lyByAiNkXEG4H1bfy8Z0bKbun37en2YeDQhuMOSbftIyJW\nRcRgRAxOmzatjRCsXeM1ZDfwaupCBQ860AdYPjw6vnyaXYMf95N6RFzZxs9bByxNHy8FbmrY/t50\nJO0JwE8bynjWA1lGx7qB1859wGPAYETcFhEfBs4GPi3pujbf031AQTXrAzw6vnyyrkXfsrQDOBmY\nKmkrcAHJiPwbJJ0LPAmckR5+C8no2c0ky2K+r1tx2b5aWTsefIOYGjlp9If8iNgEvFHSB5q92H1A\nefj+EdU04TS5ovMUmc6Yv3L9mNfWBvr7+N6KyVRjrRt6NU2uDNwHdIb7gPLo2DQ5qwcPnjOrN/cB\n1eQEbx48Z1Zz7gOqyQm+BpoNnvHgObPq8gDa+uraIDsrhiyDZzx4zqyaPIC23pzgK26i1ec8Otas\n2rK2f3AfUEUu0VecB8+Y1Zfbf735DL7ivLykWXU1u7uj23+9+Qy+xDx4xqy+stzd0e2/3pzgSyrr\nrVu9vKRZNWW5u6Pbf725RF9SHjxjVm9Zr6+7/deXE3xJefCMWXU1u7YOvr5uzblEX1DNrq975Smz\nasp6+c3X160ZJ/gC8uAZs/rKcm0dfH3dmnOJvoCyXF/3ylNm5dSs/N7K5TdfX7eJOMH3WJZrax48\nY1ZNWZaO9bV16xSX6Hso67U1X183q6Ys5XdffrNOcYLvoazX1tzAzcony8JTWapzvrZuneISfYd0\nuvQOvr5uVhZZ79qWtfzuy2/WCU7wHdDpxj3yOjdws/xl+fCedeGp5Qvn7tVXgKtz1j0u0WfQrPTm\n0rtZNWUdN9NKdc7ld+uVWp/BZ/lknuXs3KV3s3Jq1gdkPTN3dc6KqFAJXtIi4FJgCnBVRKxs972a\nNdysZfUsDdyN26wzOtUH9PrDu0vvVkSFKdFLmgJcBrwNOBI4S9KR7bxXlrJa1rJ6lgbu0rvZ5HWq\nD8haVs/SB2SdsurSuxVRkc7gjwM2R8TjAJKuBxYDD7f6RlnOurN+Ms9ydu7Su1lHdKQPyFpWz/rh\nPeuZuatzVjRFSvADwFMNz7cCx48+SNIyYBnAzJkzx3yjLA03a1k9awN34zabtI70Af7wbpYoTIk+\nq4hYFRGDETE4bdq0MY/JUlbLWlZ36c2sWJr1AVnL6q30Ad9bsYAnVv4W31uxwG3fSqNIZ/DDwKEN\nzw9Jt7Usy1l3K5/MfXZu1hMd6QNaqbqBz86tuoqU4O8F5kiaTdKozwTe084bZW24TtxmhdKRPsAf\n3s0ShUnwEbFH0vnArSRTZFZHxEPtvp8brlm5dLIPcPs3A0VE3jG0TdIO4Mkmh00FftyDcLrF8eer\niPEfFhFjD0CpmQx9QBH/fq1w/PkqYvyZ23+pE3wWkoYiYjDvONrl+PNV9vjrrux/P8efr7LHX7pR\n9GZmZtacE7yZmVkF1SHBr8o7gEly/Pkqe/x1V/a/n+PPV6njr/w1eDMzszqqwxm8mZlZ7TjBm5mZ\nVVBlE7ykRZIelbRZ0oq842mVpC2SHpC0SdJQ3vE0I2m1pO2SHmzYdpCk2yU9ln4/MM8YJzJO/J+U\nNJz+DTZJOi3PGK017gN6y31A8VQywXfy3vI5e3NEHF2SeZjXAItGbVsB3BERc4A70udFdQ37xg9w\nSfo3ODoibulxTNYm9wG5uAb3AYVSyQRPw32lI+IFYOS+0tYlEXE3sHPU5sXAmvTxGmBJT4NqwTjx\nW3m5D+gx9wHFU9UEP9Z9pcu2MHUAt0nakN7/uoymR8S29PHTwPQ8g2nT+ZLuT8t3hS0v2j7cBxSD\n+4AcVTXBV8GbIuIYkhLjeZJOyjugyYhkPmbZ5mReDhwBHA1sAz6fbzhWM+4D8lfqPqCqCb5j95bP\nS0QMp9+3AzeSlBzL5hlJMwDS79tzjqclEfFMRLwYEb8ArqScf4O6ch9QDO4DclTVBP/SfaUl7U9y\nX+l1OceUmaRflvSqkcfAqcCDE7+qkNYBS9PHS4GbcoylZSMdU+p3KOffoK7cBxSD+4AcFeZ+8J3U\n6XvL52A6cKMkSP5GX42I7+Qb0sQkXQecDEyVtBW4AFgJ3CDpXJJbep6RX4QTGyf+kyUdTVJW3AJ8\nMLcArSXuA3rPfUDxeKlaMzOzCqpqid7MzKzWnODNzMwqyAnezMysgpzgzczMKsgJ3szMrIKc4M3M\nzCrICd7MzKyCnOCtJZJmSXpE0pWSHpJ0m6S+vOMys+5z+y8XJ3hrxxzgsoh4HbALeGfO8ZhZ77j9\nl4QTvLXjiYjYlD7eAMzKMRYz6y23/5Jwgrd2PN/w+EUqek8DMxuT239JOMGbmZlVkBO8mZlZBflu\ncmZmZhXkM3gzM7MKcoI3MzOrICd4MzOzCnKCNzMzqyAneDMzswpygjczM6sgJ3gzM7MKcoI3MzOr\nICd4MzOzCnKCNzMzq6BS3wVo6tSpMWvWrLzDMOupDRs2/DgipuUdRxG4D7C6aaX9lzrBz5o1i6Gh\nobzDMOspSU/mHUNRuA+wumml/btEb2ZmVkGlPoM3q4q1G4e5+NZH+dGu3Rzc38fyhXNZMm8g77DM\nrEe60Qc4wZvlbO3GYT7xzQfY/fMXARjetZtPfPMBACd5sxroVh/gEr1Zzi6+9dGXGvaI3T9/kYtv\nfTSniMysl7rVBzjBm+XsR7t2t7TdzKqlW32AE7xZzg7u72tpu5lVS7f6ACd4s5wtXziXvv2m7LWt\nb78pLF84N6eIzKyXutUHeJCdWZc1Gx078tij6M3qqVt9gBO8WRdlHR27ZN6AE7pZjXWjD3CJ3qyL\nPELezPLiBG/WRR4hb2Z5cYI36yKPkDezvDjBm7Vp7cZh5q9cz+wVNzN/5XrWbhze5xiPkDezvHiQ\nnVkbWhk8Bx4hb2a95wRv1oaJBs+NTt5VHyEvqR+4Cng9EMD7gUeBrwGzgC3AGRHxrCQBlwKnAc8B\n50TEfen7LAX+W/q2n46INT38NcwqxyV6szZ48NxeLgW+ExGvBd4APAKsAO6IiDnAHelzgLcBc9Kv\nZcDlAJIOAi4AjgeOAy6QdGAvfwmzqnGCN2uDB88lJL0aOAm4GiAiXoiIXcBiYOQMfA2wJH28GLg2\nEvcA/ZJmAAuB2yNiZ0Q8C9wOLOrhr2JWOU7wZm3w4LmXzAZ2AH8paaOkqyT9MjA9IralxzwNTE8f\nDwBPNbx+a7ptvO37kLRM0pCkoR07dnTwVzHrvCyDcbvF1+DNxuDlZTN7BXAM8IcR8QNJl/JyOR6A\niAhJ0akfGBGrgFUAg4ODHXtfs07r1n3es3KCNxvFy8u2ZCuwNSJ+kD7/BkmCf0bSjIjYlpbgt6f7\nh4FDG15/SLptGDh51Pa7uhi3Wde1Mhi3G1yiNxvFy8tmFxFPA09JGrk2cQrwMLAOWJpuWwrclD5e\nB7xXiROAn6al/FuBUyUdmA6uOzXdZlZaeQ/G9Rm82Sh5N8oS+kPgryTtDzwOvI/k5OEGSecCTwJn\npMfeQjJFbjPJNLn3AUTETkmfAu5Nj7swInb27lcw67yD+/sYHqPf6NVgXCd4s1HybpRlExGbgMEx\ndp0yxrEBnDfO+6wGVnc2OrP8LF84d6/LfdDbwbgu0ZuN4hHyZtYJS+YNcNHpRzHQ34eAgf4+Ljr9\nqJ6N3en5Gbyk1cDbge0R8fp02yeBD5BMtwH4k4i4pdexWfU1Gx0PHiFvZp2T52DcPEr01wB/Dlw7\navslEfG53odjddHKlBWPkDezsut5iT4i7gY8eMZ6zqPjzaxOinQN/nxJ90taPdEa1F7Fytrl0fFm\nVidFSfCXA0cARwPbgM+Pd2BErIqIwYgYnDZtWq/iswrw+vFmVieFSPAR8UxEvBgRvwCuJLmblFlH\neXS8mdVJIebBjyxpmT79HeDBPOOx8vHoeDOzveUxTe46kjWnp0raSnIP6JMlHQ0EsAX4YK/jsvLy\n6Hgzy0OWE4s89TzBR8RZY2y+utdxWHXkfUMHM6ufvO8Ul0UhrsGbTYZHx5tZr5Vh2q0TvJWeR8eb\nWa+V4cTCCd5Kz6PjzazXynBi4QRvhbd24zDzV65n9oqbmb9yPWs3Du+1P+8bOphZ/ZThxKIQ0+TM\nxpN1IItHx5tZL5Vh2q0TvBWaR8j3hqQDSO7yeCJwMLCbZD2KmyPioTxjMyuqop9YTCrBS/ol4A00\ndAgRsb0TgZlBOQaylJ2k/0mS3O8CfgBsBw4AfgNYmSb/j0bE/bkFaWYtayvBSzoC+DjwFuAxkvu4\nHwD8hqTngCuANenSs2ZtO7i/j+ExknmRBrJUwA8j4oJx9n1B0q8DM3sZkJlNXruD7D4N/G/giIhY\nGBH/OSJ+NyJ+E3gH8Grg7E4FafVVhoEsZRcRNzfZvz0ihnoVj5l1Rltn8OOsRjeybzvwxbYjstrw\n+vHFIuk3gOXAYTT0DRGxILegzKxtkx5kJ+mNwCz27hCunez7WrV5/fhC+jrwFyR3dHyxybFmVnCT\nHWT3FZL7uG/i5Q4hACd4m5BHxxfSnoi4PO8gzPJW9JvIZDXZM/hB4MiIiE4EY/Xh0fGF9C1JfwDc\nCDw/sjEidjZ7oaQpwBAwHBFvlzQbuB74NWADcHZEvCDplSQnAMcCPwHeHRFb0vf4BHAuycnCH0XE\nrZ385cyyKMNNZLKa7Ep2DwL/vhOBWL2UYZnHGlpKcg3+70mS8gaSpJ3Fh4BHGp5/BrgkIl4DPEuS\nuEm/P5tuvyQ9DklHAmcCrwMWAV9OPzSY9VQZbiKT1WQT/FTgYUm3Slo38tWJwKzaPDq+eCJi9hhf\nhzd7naRDgN8CrkqfC1gAfCM9ZA2wJH28OH1Ouv+U9PjFwPUR8XxEPAFsBo7r1O9mllWVqouTLdF/\nshNBWPU0u4bl0fHFIelNEfF3E+z/VWBmRDw4ziFfBD4GvCp9/mvArojYkz7fCoz8YQeApwAiYo+k\nn6bHDwD3NLxn42tGx7MMWAYwc6an51tnVWntjXYXulEk/rbZMe2HZmXl9eNL552SPgt8h6QsP7Jw\n1WuAN5NMm/voWC+U9HZge0RskHRyL4KNiFXAKoDBwUH3MdZRyxfO3av/gvJWF9s9g79T0l8DN0XE\nP49slLQ/8CaSa3l3AtdMOkIrHY+QL5eI+Iikg4B3Au8CZpAsPf0IcMVEZ/fAfOAdkk4j+VDwq8Cl\nQL+kV6Rn8YcAI7cAHAYOBbZKegXJolg/adg+ovE1Zj1Tpepiuwl+EfB+4Lp0tOwuoI/kmv5twBcj\nYmNnQrSyqdI1rDqQNCsdyX5l+pVZRHwC+ET6PicD/zUifk/S14HfJRlJvxS4KX3JuvT599P96yMi\n0rE7X5X0BZJ7W8wBfjjJX82sLVWpLra7kt2/AV8mGem6H8lgu90RsauTwVk5VekaVk18V9JVwOca\nrptP1seB6yV9GtgIXJ1uvxr4iqTNwE6SkfNExEOSbgAeBvYA50WEF9sxm4S2RtFLukXSLICI+HlE\nbHNytxEeIV8684DpwAZJJ7b7JhFxV0S8PX38eEQcFxGviYh3RcTz6fZ/S5+/Jt3/eMPr/ywijoiI\nuRHx7cn+UmZ1126J/i+B2yStAT4bET9v5cWSVpPcnnJ7RLw+3XYQ8DWSZW+3AGdExLNtxmdd4vXj\nqycifgZ8RNKxwB2StgK/AJTsjt/MNUAza0u7JfqvS/o28N+BoXTJ2l807P9Ck7e4Bvhz9l7SdgVw\nR0SslLQiff7xduKz7vD68dUlaQHJ4LirgMtoaM9mVVGVJWizmsw8+BeA/we8kmT+a+YOISLuHinx\nN1gMnJw+XgPchRN8oXh0fDVJup5k1Pp7IuKBvOMx64YqLUGbVbvz4BcBXyAZEXtMRDzXgVimR8S2\n9PHTJNcEx/rZXuQiJx4dX1kPA2d53QqrsjqeoLS7VO2fAu+KiBUdSu57STuaMTubiFgVEYMRMTht\n2rRO/2ibgNePr6wZJAPsrpd0jiTfX8Iqp44nKG0l+Ig4MSIeApB0oKTXSTpc0mTWtn9G0oz0PWcA\n2yfxXtYFHh1fTRHx+xFxDMnS0wcC10j6vqT/Jekk3/TFqqCOJyjtTpN7taQ/kfQAyfrRVwA3AE9K\n+rqkN7fxtiMLYMDeC2NYD6zdOMz8leuZveJm5q9cz9qN+y4itmTeABedfhQD/X0IGOjv46LTj6ps\neatuIuIfIuKSiFhEcrOYvyNZ2e4H+UZmNnl1PEFpd5DdN0hGwJ84ev57OtXmbEmHR8TVY71Y0nUk\nA+qmplNyLgBWAjdIOhd4EjijzdisRR4dbyMkHUiyktxu4DsRcUvOIZl1RB2n77Y7Te6tE+wbuY/0\nRK8/a5xdp7QTj01OHQef2MskvRo4DzgL2J+XbzYzXdI9wJcj4s4cQzTriLqdoEz2drFIGiC529RL\n7xURd0/2fa136jj4xPYyqYqcmRXTpBK8pM8A7yaZZjNyChiAE3yJeO34eptsRc7MimmyZ/BLgLkj\n60xbOVXp/sc2Oa7IWVnVbZW6LCab4B8H9gOc4Aus2X/8Og4+sX25ImdlVcdV6rKYbIJ/Dtgk6Q4a\nknxE/NEk39c6JOt//LoNPrExuSJnpeSBwmObbIJfl35ZQfk/vrXAFTkrJQ8UHtukEnxErOlUINYd\n/o9vLXBFzkrJA4XH1u5Kdt+S9NuS9htj3+GSLpT0/smHZ5NVx+UZrW3rgE8Bf08yct4j6K0U6rhK\nXRbtnsF/APhj4IuSdvLywhizgc3An0eEl5otAI+Qt6xckbOy8kDhsbWb4PePiI8BH0vv6z6DZGnL\nfwSOjYj/05nwbCJZpoX4P741I+lbwCqSpWl/Pmrf4cA5wJaIWJ1DeGaZeKDwvtpN8HdJ+gvg8xGx\nBdgiaTpJJ/FaYLBD8dk4vH68dZArclZYnt/evnZv73oscATJgJwFkj4E/BD4PnBcp4Kz8U00Ot6s\nRftHxMci4giSu8d9iiThvw640Mnd8jJyIjO8azfByycyY93t0vbV7v3gn42IDwJXAd8FlgPzI+Ky\niPhFJwO0sXl0vHXQXZI+JmlKRGyJiO8D20gqcpfkHJvVmE9kJqfdUfT9kq4A3gcsIrlZxbclLehk\ncDY+j463Dmq7IifpUEl3SnpY0kPpa5F0kKTbJT2Wfj8w3S5JX5K0WdL9ko5peK+l6fGPSVratd/W\nSsMnMpPTbon+PuAxYDAibouIDwNnA59O7/VuXeZpIdYpk6zI7QE+GhFHAicA50k6ElgB3BERc4A7\n0ucAbwPmpF/LgMsh+UAAXAAcT/Kh4oKRDwVWXz6RmZx2E/xJEfG5iNgzsiEiNkXEG4H1nQmt3tZu\nHGb+yvXMXnEz81eu3+ea05J5A1x0+lEM9PchYKC/j4tOP8qDT6xlk6nIRcS2iLgvffwz4BFgAFgM\njEy7W0OyDC7p9msjcQ/QL2kGsBC4PSJ2RsSzwO1pLFZjPpGZnLZG0UfE1gn2Xdl+OAZeP9567j7g\ny8B56Yf22yQdDXxZ0pMRcVaWN0mnzM4DfgBMj4ht6a6ngenp4wHgqYaXbU23jbd9rJ+zjOTsn5kz\nZ2YJzUrK03wnZ7Jr0VsXeP1467GTRn9oj4hNwBslfSDLG0j6FeCvgQ9HxL9IanyvkBSdCjYiVpEM\nAGRwcLBj72vF5BOZ9jnBF5AHllgvTbYily5Z/dfAX0XEN9PNz0iaERHb0hL89nT7MHBow8sPSbcN\nAyeP2n5X1t/Byslz3Lur3Wvw1kUeWGJloeRU/WrgkYj4QsOudcDISPilwE0N29+bjqY/AfhpWsq/\nFThV0oHp4LpT021WUZ7j3n2FS/CStkh6QNImSUN5x5MHDyyxEplPMoNmQdpmN0k6DVgJvFXSY8Bb\n0ucAt5DclnYzcCXwBwARsZNkgZ17068L021WUZ7j3n1FLdG/OSJ+nHcQ3eD1461KIuLvAI2z+5Qx\njg/gvHHeazXg9e5rwpciu6+oCb6SvH68mVnC93DvvsKV6IEgmaazIZ0OUxkuSZmZJXwpsvuKeAb/\npogYlvTrwO2S/iEi7h7ZWeY5sC5JmVkd+FJkMRQuwUfEcPp9u6QbSZatvLthf2nnwLokZWZV50uR\nxVGoEr2kX5b0qpHHJFNlHsw3qs5xScrMqs6XIoujaGfw04Eb01WwXgF8NSK+k29I2bgkZWbmS5FF\nUqgEHxGPA2/IO45WuSRlZpbwpcjiKFSJvqxckjKzOmh2l0vwpcgiKdQZfFm5JGVmVdfKXS7BlyKL\nwAm+A1ySMrOqa+Uul74UWQwu0XeAS1JmVnWuVJaPz+AzaDZC3iUpMyu7Zv2cK5Xl4wTfRCvXnZzQ\nzayMsvRzyxfO3esYcKWy6Fyib8Ij5M2s6rL0c0vmDXDR6Ucx0N+HgIH+Pi46/Sif2BSYz+Cb8HUn\nMyuzLItwZe3nXKksF5/BNzHe9SVfdzKzohspvQ/v2k3wcul99Px193PV5ATfhEfIm1lZZb3E6H6u\nmmpdovf68WZWZs36sFZK7+B+rmpqm+C9fryZlVmWPqyVqW3u56qntiV6j443s6LKsuZ7lj7Mpfd6\nq+0ZvEfHm1kRZa0uZunDXHqvt9omeK/KZGadlGVMT5bjsq75nrUPc+m9viqb4Js1Iq/KZFY8khYB\nlwJTgKsiYmU779OpZNvKMVnOurMcl7W66D7MmqnkNfgscz+9KpNZsUiaAlwGvA04EjhL0pGtvk/W\nud9Zjsv6XlnH9GQ5LuucdPdh1kwlz+CzlrhcujIrlOOAzRHxOICk64HFwMOtvEnW9p/luKzvlfWs\nO8txrZyZuw+ziVTyDN4D6MxKaQB4quH51nTbXiQtkzQkaWjHjh37vEknk23W98p61p3lOJ+ZW6dU\nMsF72UWz6oqIVRExGBGD06ZN22d/J5Nt1vfKOh0t63FL5g3wvRULeGLlb/G9FQuc3K0tlUzwnvtp\nVkrDwKENzw9Jt7Wkk8m2lYSc5azbZ+fWS5W8Bu+5n2aldC8wR9JsksR+JvCeVt8ka/vPclwrfUnW\n6+G+bm69oojIO4a2DQ4OxtDQUN5hmPWUpA0RMZh3HN0g6TTgiyTT5FZHxJ9NdLz7AKubVtp/Jc/g\nzaycIuIW4Ja84zCrglKfwUvaATzZ5LCpwI97EE63OP58FTH+wyJi39FlNZShDyji368Vjj9fRYw/\nc/svdYLPQtJQmcuZjj9fZY+/7sr+93P8+Sp7/JUcRW9mZlZ3TvBmZmYVVIcEvyrvACbJ8eer7PHX\nXdn/fo4/X6WOv/LX4M3MzOqoDmfwZmZmteMEb2ZmVkGVTfCSFkl6VNJmSSvyjqdVkrZIekDSJkmF\nX6pL0mpJ2yU92LDtIEm3S3os/X5gnjFOZJz4PylpOP0bbEpXWbOScB/QW+4DiqeSCV7SFOAy4G3A\nkcBZko7MN6q2vDkiji7JPMxrgEWjtq0A7oiIOcAd6fOiuoZ94we4JP0bHJ2usmYl4D4gF9fgPqBQ\nKpnggeOAzRHxeES8AOwhmS4AAAIySURBVFwPLM45pkqLiLuBnaM2LwbWpI/XAEt6GlQLxonfyst9\nQI+5Dyieqib4AeCphudb021lEsBtkjZIWpZ3MG2aHhHb0sdPA9PzDKZN50u6Py3fFba8aPtwH1AM\n7gNyVNUEXwVviohjSEqM50k6Ke+AJiOS+Zhlm5N5OXAEcDSwDfh8vuFYzbgPyF+p+4CqJvhh4NCG\n54ek20ojIobT79uBG0lKjmXzjKQZAOn37TnH05KIeCYiXoyIXwBXUs6/QV25DygG9wE5qmqCvxeY\nI2m2pP2BM4F1OceUmaRflvSqkcfAqcCDE7+qkNYBS9PHS4GbcoylZSMdU+p3KOffoK7cBxSD+4Ac\nVfJ+8BGxR9L5wK3AFGB1RDyUc1itmA7cKAmSv9FXI+I7+YY0MUnXAScDUyVtBS4AVgI3SDqX5Jae\nZ+QX4cTGif9kSUeTlBW3AB/MLUBrifuA3nMfUDxeqtbMzKyCqlqiNzMzqzUneDMzswpygjczM6sg\nJ3gzM7MKcoI3MzOrICd4MzOzCnKCNzMzqyAneGuJpFmSHpF0paSHJN0mqS/vuMys+9z+y8UJ3tox\nB7gsIl4H7ALemXM8ZtY7bv8l4QRv7XgiIjaljzcAs3KMxcx6y+2/JJzgrR3PNzx+kYre08DMxuT2\nXxJO8GZmZhXkBG9mZlZBvpucmZlZBfkM3szMrIKc4M3MzCrICd7MzKyCnODNzMwqyAnezMysgpzg\nzczMKsgJ3szMrIL+f/ZP+WculP42AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 576x576 with 6 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# You can easily visualize a sequence by plotting it.\n",
    "\n",
    "fig,ax = plt.subplots(3,2, figsize=(8,8))\n",
    "\n",
    "ax[0,0].plot(X,'o');   ax[0,0].set_xlabel(\"n\"); ax[0,0].set_ylabel(\"X(n)\");\n",
    "ax[0,1].plot(Y,'o');   ax[0,1].set_xlabel(\"n\"); ax[0,1].set_ylabel(\"Y(n)\");\n",
    "ax[1,0].plot(X+Y,'o'); ax[1,0].set_xlabel(\"n\"); ax[1,0].set_ylabel(\"X(n)+Y(n)\");\n",
    "ax[1,1].plot(X-Y,'o'); ax[1,1].set_xlabel(\"n\"); ax[1,1].set_ylabel(\"X(n)+-Y(n)\");\n",
    "ax[2,0].plot(X*Y,'o'); ax[2,0].set_xlabel(\"n\"); ax[2,0].set_ylabel(\"X(n)Y(n)\");\n",
    "ax[2,1].plot(X/Y,'o'); ax[2,1].set_xlabel(\"n\"); ax[2,1].set_ylabel(\"X(n)/Y(n)\");\n",
    "plt.subplots_adjust(hspace=0.35, wspace=0.35)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limits of sequences\n",
    "\n",
    "**Def:** Let $X = (x_n)$ be a sequence. $x$ is a **limit** of $X$ if for every $\\varepsilon > 0$ there exists $K(\\varepsilon) \\in \\mathbb{N}$ such that for all $n \\geq K(\\varepsilon)$, \n",
    "\n",
    "  $$\n",
    "  x - \\varepsilon < x_n < x + \\varepsilon .\n",
    "  $$\n",
    "  \n",
    "In this case we write $\\lim x_n = x$ and say that $x_n$ converges to $x$.\n",
    "  \n",
    "**Example:** Show that the limit of the sequence $x_n = \\frac{1}{n}$ is 0. \n",
    "\n",
    "Suppose $\\varepsilon > 0$ is chosen. Then we need \n",
    "\n",
    "$$\n",
    "\\frac{1}{n} < \\varepsilon .\n",
    "$$\n",
    "\n",
    "Multiplying by $n/\\varepsilon$ (which is positive) gives\n",
    "\n",
    "$$\n",
    "\\frac{1}{\\varepsilon} < n .\n",
    "$$\n",
    "\n",
    "Thus, we choose $K(\\varepsilon) = \\left \\lceil \\frac{1}{\\varepsilon} \\right \\rceil$ to satisfy the definition."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An equivalent way to show that a sequence converges to a limit point is to use the following theorem, which can be shown by using the above definition.\n",
    "\n",
    "**Thm 1:** Let $(a_n)$ and $(x_n)$ be sequences with $\\lim a_n = 0$ and let $x \\in \\mathbb{R}$. If for some $C > 0$ and some $m > 0$ it is the case that\n",
    "\n",
    "$$\n",
    "\\left |x_n - x \\right | \\leq C\\left |a_n\\right |\n",
    "$$\n",
    "\n",
    "for all $n > m$, then $\\lim x_n = x$. \n",
    "\n",
    "**Example:** Show that the limit of the sequence $x_n = \\frac{1}{2^n}$ is 0. \n",
    "\n",
    "First, note that $0 < n < 2^n$ when $n > 0$. Then, dividing by $n2^n$, \n",
    "\n",
    "  $$\n",
    "  0 < \\frac{1}{2^n} < \\frac{1}{n}.\n",
    "  $$\n",
    "  \n",
    "Then, subtracting 0 and taking absolute values gives \n",
    "\n",
    "  $$\n",
    "  \\left | \\frac{1}{2^n} - 0 \\right | \\leq \\frac{1}{n} .\n",
    "  $$\n",
    "  \n",
    "Thus, using the theorem above with $C = 1$ and $m=1$ we can conclude that\n",
    "\n",
    "$$\\lim \\frac{1}{2^n} = 0$$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:31.161175Z",
     "start_time": "2019-10-09T22:14:30.947543Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# You can see the above graphically.\n",
    "\n",
    "A = [1/n for n in range(1,20)]\n",
    "X = [1/(2**n) for n in range(1,20)]\n",
    "\n",
    "plt.plot(A,\"o\",label=\"a_n\")\n",
    "plt.plot(X,\"o\",label=\"x_n\")\n",
    "plt.xlabel(\"n\")\n",
    "plt.legend();\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example:** Show that $\\lim \\frac{1}{1+nr} = 0$ when $r>0$. \n",
    "\n",
    "Since $r > 0$,\n",
    "\n",
    "$$\n",
    "0 < nr < 1+nr.\n",
    "$$\n",
    "\n",
    "Dividing by $nr(1+nr)$ gives\n",
    "\n",
    "$$\n",
    "0 < \\frac{1}{1+nr} < \\frac{1}{nr}\n",
    "$$\n",
    "\n",
    "implying that\n",
    "\n",
    "$$\n",
    "\\left | \\frac{1}{1+nr} \\right | \\leq \\frac{1}{r} \\left | \\frac{1}{n} \\right |.\n",
    "$$\n",
    "\n",
    "Thus, the theorem above applies with $C = \\frac{1}{r}$ and $m = 1$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:32.323907Z",
     "start_time": "2019-10-09T22:14:32.112631Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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OBW40s3N7dfsOsMndS4AbgJ+ku9DQWnAH5Ob3bMvNj7WLiAQkkTv3i4GD7n7I3U8BG4Fr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\nhbuISAQp3EVEIkjhLiISQQp3EZEI+v+xDbdSSyIouwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# You can see the above graphically.\n",
    "\n",
    "r = 0.6\n",
    "A = [1/n for n in range(1,10)]\n",
    "X = [1/(1 + r*n) for n in range(1,10)]\n",
    "\n",
    "plt.plot(A,\"o\",label=\"a_n\")\n",
    "plt.plot((1/r)*np.array(A),\"o\",label=\"(1/r)a_n\")\n",
    "plt.plot(X,\"o\",label=\"x_n\")\n",
    "plt.xlabel(\"n\")\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limits of Combined sequences\n",
    "\n",
    "**Thm 2:** If $\\lim x_n = L$ and $\\lim y_n = M$ then \n",
    "\n",
    "$$\n",
    "\\lim x_n + y_n = L + M\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\lim a x_n = a L\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\lim x_n y_n = L M\n",
    "$$\n",
    "\n",
    "and when $y_n \\neq 0$,\n",
    "\n",
    "$$\n",
    "\\lim \\frac{x_n}{y_n} = \\frac{L}{M}.\n",
    "$$\n",
    "\n",
    "**Proof:** For $\\lim x_n + y_n = L + M$, note that by the [triangle inequality](https://en.wikipedia.org/wiki/Triangle_inequality),\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\left | (x_n + y_n ) - ( L+M ) \\right | & = & \\left | (x_n-L) + (y_n - M) \\right | \\\\\n",
    "                                        & \\leq & \\left |(x_n - L) \\right | + \\left |(y_n - M ) \\right |, \n",
    "\\end{eqnarray}\n",
    "\n",
    "If $\\varepsilon > 0$ is given, then there exist natural numbers $K_1$ and $K_2$ such that if $n > \\max(K_1,K_2)$ then \n",
    "\n",
    "$$\\left |(x_n - L) \\right | < \\frac{\\varepsilon}{2}$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\\left |(y_n - M) \\right | < \\frac{\\varepsilon}{2}$$\n",
    "\n",
    "so that \n",
    "\n",
    "$$\n",
    "\\left |(x_n - L) \\right | + \\left |(y_n - M ) \\right | < \\frac{\\varepsilon}{2} + \\frac{\\varepsilon}{2} = \\varepsilon .\n",
    "$$\n",
    "\n",
    "The other properties are demonstrated similarly.\n",
    "\n",
    "&square;\n",
    "\n",
    "**Example**: $\\lim \\frac{1}{n} + \\frac{1}{2^n} = 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Limits of Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "**Def:** Let $f : A \\rightarrow \\mathbb{R}$ and $c \\in A$ be a cluster point (which we won't define here, but it is just non-isolated point). Then\n",
    "\n",
    "$$\n",
    "\\lim_{x \\rightarrow c} f(x) = L\n",
    "$$\n",
    "\n",
    "if and only if for every $\\varepsilon > 0$, there exists a $\\delta > 0$ such that if $x \\in A$ and\n",
    "\n",
    "$$0 < |x-c| < \\delta$$\n",
    "\n",
    "then \n",
    "\n",
    "$$|f(x)-L| < \\epsilon. $$\n",
    "\n",
    "**Example:** Show $\\lim_{x \\rightarrow c} x = c$. \n",
    "\n",
    "Given $\\varepsilon$ simply choose $\\delta = \\varepsilon$. Then \n",
    "\n",
    "$$\n",
    "0 < |x-c| < \\varepsilon\n",
    "$$\n",
    "\n",
    "is equivalent to \n",
    "\n",
    "$$\n",
    "0 < |x-c| < \\delta .\n",
    "$$\n",
    "\n",
    "<hr>\n",
    "\n",
    "The following is an alternative way to define the limit. It is shown here as a theorem, meaning that you can prove it by assume the definition above of a limit. Or you can \n",
    "\n",
    "**Thm 3:** Let $f:A \\rightarrow \\mathbb{N}$. Then $\\lim_{x \\rightarrow c} f(x) = L$ if and only if for every sequence $(x_n) \\in A$ that converges to $c$ such that $x_n \\neq c$, the sequence $f(x_n)$ converges to $L$.\n",
    "\n",
    "One use of this theorem to show a limit does *not* exist, we have to find just one sequence that does not satisfy the above property. \n",
    "\n",
    "**Example**: $\\lim_{x \\rightarrow 0} \\frac{1}{x}$ does not exist. \n",
    "\n",
    "To show this, let $x_n = 1/n$. We have already shown this sequence converges to $0$, which is also is not one of the elements of the sequence. However, if $f(n) = \\frac{1}{x}$ then $f(x_n) = 1 / ( 1 / n ) = n$, which does not converge.\n",
    "\n",
    "<hr>\n",
    "\n",
    "The above theorem can also be used to show the various formulas about limits we use all the time.\n",
    "\n",
    "**Thm 4:** Say $\\lim_{x \\rightarrow c}f(x) = L$ and $\\lim_{x \\rightarrow c}f(x) = M$, then\n",
    "\n",
    "$$\n",
    "\\lim_{x \\rightarrow c}f(x) + g(x) = L + M\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\lim_{x \\rightarrow c} a f(x) = a L\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\lim_{x \\rightarrow c}f(x) g(x) = L M\n",
    "$$\n",
    "\n",
    "and, when $M \\neq 0$, \n",
    "\n",
    "$$\n",
    "\\lim_{x \\rightarrow c}f(x) / g(x) = L / M .\n",
    "$$\n",
    "\n",
    "**Proof:** Let $x_n$ be any sequence that converges to $c$ but does not contain $c$. Then by Thm 3, $\\lim f(x_n) = L$ and $\\lim g(x_n) = M$. Thus, by Thm 2, $\\lim f(x_n) + g(x_n) = L+M$. Similar arguments hold for the remaining statements. &square;\n",
    "\n",
    "**Example:** $\\lim_{x \\rightarrow 1} \\frac{x^2 + 2}{x^3 + 1} = \\frac{3}{2}.$\n",
    "\n",
    "In general, limits of polynomials and quotients of polynomials can be computed this way.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limits with Sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:41.691013Z",
     "start_time": "2019-10-09T22:14:41.685845Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{3}{2}$$"
      ],
      "text/plain": [
       "3/2"
      ]
     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sm.symbols(\"x\")\n",
    "\n",
    "sm.limit((x**2 + 2)/(x**2+1),x,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:42.167786Z",
     "start_time": "2019-10-09T22:14:42.162964Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 172,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.limit(x**(3/2),x,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Continuous Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Def:** A function $f : A \\rightarrow \\mathbb{R}$ is continuous at $c$ if and only if for any $\\varepsilon$ there exists $\\delta$ such that for all $x \\in A$, if $|x-c| < \\delta$ then $|f(x) - f(c)| < \\varepsilon$.\n",
    "\n",
    "**Equivalent Def:** A function $f : A \\rightarrow \\mathbb{R}$ is continuous if and only if \n",
    "\n",
    "$$\n",
    "f(c) = \\lim_{x \\rightarrow c} f(x)\n",
    "$$\n",
    "\n",
    "**Example:** If $f(x) = x$, then $\\lim_{x \\rightarrow c} f(x) = c$. Also, $f(c) = c$. Thus, $f(x)$ is continuous at all points $c$. \n",
    "\n",
    "<hr>\n",
    "\n",
    "**Example:** If $f(x) = 1/x$ then if $c > 0$,  $\\lim_{x \\rightarrow c} f(x) = 1/c$. Also, $f(c) = 1/c$. So $f(x)$ is continuous at points $c > 0$. \n",
    "\n",
    "<hr>\n",
    "\n",
    "**Equivalent Def:** A function $f : A \\rightarrow \\mathbb{R}$ is continuous at $c$ if and only if for any sequence $(x_n)$ that converges to a value $c$, the sequence $(f(x_n))$ converges to $f(c)$. That is,\n",
    "\n",
    "$$\n",
    "\\lim x_n = c \\Rightarrow \\lim f(x_n) = f(c) .\n",
    "$$\n",
    "\n",
    "**Non-Convergence Criterion:** Find *some* sequence $(x_n)$ that converges to $c$ where $f(x_n)$ does not converge to $f(c)$. \n",
    "\n",
    "**Example:** $f(x) = 1/x$ is not continuous at $0$. To see this, let $(x_n)$ be defined by $x_n = 1/n$. Then as we have shown, $\\lim x_n = 0$. However, the sequence $(f(x_n)) = 1/(1/n) = n$ does not converge. \n",
    "\n",
    "<hr>\n",
    "\n",
    "**Example:** Dirichlet's Function (1829). Define $f$ by\n",
    "\n",
    "$$\n",
    "f(x) = \\left \\{\n",
    "\\begin{array}{ll}\n",
    "1 & \\mathrm{if}\\;x\\;\\mathrm{is\\;rational} \\\\\n",
    "0 & \\mathrm{otherwise}. \n",
    "\\end{array}\n",
    "\\right .\n",
    "$$\n",
    "\n",
    "Then $f$ is discontinuous everywhere. \n",
    "\n",
    "It turns out a fundamental property of the real numbers is the following. If $x<y$, then there exists a rational number $z$ with $x<z<y$. In addition, there exists an irrational number $w$ with $x<w<y$. \n",
    "\n",
    "Thus, if $c$ is rational, we can come up with an irrational sequence $(z_n)$ that converges to $c$. Then $f(c) = 1$ but $(f(z_n)) = 0$. \n",
    "\n",
    "Conversely, if $c$ is irrational, choose a sequence $(w_n)$ of rational numbers that converges to $c$. \n",
    "\n",
    "<hr>\n",
    "\n",
    "**Example:** Thomae's Function (1875). Define $f(x)$ as follows. If $x$ is irrational, then $f(x) = 0$. If $x$ is rational, then express it in reduced form (no common factors) as $m/n$. Then define $f(m/n) = 1/n$. This is plotted below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:46.057293Z",
     "start_time": "2019-10-09T22:14:45.875548Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Mx8zMaissKUgaB1wNnAocBZwp6aiKYj8H+iLincCtwF8XFY+ZmdVX5CeajwM2RMQmAElL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\ntWNDd5ajiwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = []\n",
    "y = []\n",
    "\n",
    "for den in range(1,20):\n",
    "    for num in range(0,2*den+1):\n",
    "        x.append(num/den)\n",
    "        # Note that sympy's Rational function automatically reduces. Also, you can\n",
    "        # get the numerator and denominator with the .p and .q attributes. \n",
    "        y.append(sm.Rational(1,sm.Rational(num,den).q))\n",
    "plt.plot(x,y,'o',markersize=2)\n",
    "plt.title(\"Thomae's Function\")\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"f(x)\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function is discontinuous at rational numbers, but continuous at irrational numbers. \n",
    "\n",
    "If $c$ is rational, we can come up with an irrational sequence $(z_n)$ that converges to $c$. Then $f(c) = 1$ but $(f(z_n)) = 0$. \n",
    "\n",
    "However, if $c$ is irrational, then choose any $\\epsilon > 0$. Then there is some $n_0 \\in \\mathbb{N}$ such that $1/n_0 < \\epsilon$. There are only a finite number of rationals with denominator less than $n_0$ in $(c-1, c+1)$ (because there are only a finite number of possible denominators, and for each of those a finite number of numerators). Thus, $\\delta$ can be chosen so that if $(c-\\delta, c+\\delta)$ contains no rational numbers whose denominators are less than $n_0$. So if $|x-c| < \\delta$ then $|h(x) - h(c)| = |h(x)| < 1/n_0 < \\epsilon$. The last inequality comes from the fact that if $x$ is in $(c-\\delta, c+\\delta)$ it must have a denominator greater than $n_0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Combinations of Continuous Functions\n",
    "\n",
    "Because of the combinations for limits, combinations $f+g$, $fg$, $f-g$, and $f/g$ of continuous functions $f$ and $g$ are also continuous. In addition, $f \\circ g$ is continuous if $f$ and $g$ are."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Intermediate Value Theorem\n",
    "\n",
    "**Thm 5:** Let $I$ be an interval in $\\mathbb{R}$ and suppose $f : I \\rightarrow \\mathbb{R}$ is continuous at all points in $I$. If $a,b \\in I$ and $a<b$ and if $k \\in \\mathbb{R}$ and $f(a) < k < f(b)$ then there exists a point $c \\in I$ with $f(c) = k$. \n",
    "\n",
    "The proof follows from the definition of *continuous*.\n",
    "\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Illustration_for_the_intermediate_value_theorem.svg/1200px-Illustration_for_the_intermediate_value_theorem.svg.png\" width=300>\n",
    "\n",
    "![](https://study.com/cimages/multimages/16/intermediate-value-theorem-two.jpg)\n",
    "\n",
    "**Example:** Say $f(x) = x^2 - 2$, $a = 1$, and $b = 2$. Then $f(a) = -1$ and $f(b) = 2$. Since $-1 < 0 < 2$, it must be the case that there exists $c$ with $-1<c<2$ with $f(c) = c^2 - 2 = 0$. That is $c = \\sqrt{2}$. \n",
    "\n",
    "Interesting consequences:\n",
    "\n",
    "- If a four legged table is not resting flat, it will if you rotate it.\n",
    "- If you take a long, round trip walk in the hills, then at some point other than where you started, you will be at the same height as where you started (assuming you don't start at the lowest or highest point). \n",
    "- Same applies to pressure, temperature, etc."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Def:** If $f:I \\mathbb{R}$ function $c \\in I$, then the derivative $f'$ of $f$ is defined by \n",
    "\n",
    "$$\n",
    "f'(c) = \\lim_{x \\rightarrow c} \\frac{f(x) - f(c)}{x-c}. \n",
    "$$\n",
    "\n",
    "assuming this limit exists. \n",
    "\n",
    "**Example:** Say $f(x) = x^2$. Then\n",
    "\n",
    "\\begin{eqnarray}\n",
    "f'(c) & = & \\lim_{x \\rightarrow c} \\frac{x^2 - c^2}{x-c}  \\\\\n",
    "      & = & \\lim_{x \\rightarrow c} x+c = 2c\n",
    "\\end{eqnarray}\n",
    "\n",
    "Thus, $f'(x) = 2x$. \n",
    "\n",
    "<hr>\n",
    "\n",
    "**Thm 6:** If $f$ has a derivative at $c$ then $f$ is continous at $c$. \n",
    "\n",
    "**Proof:** We have\n",
    "\n",
    "$$\n",
    "f(x) - f(c) = \\left ( \\frac{f(x)-f(c)}{x-c} \\right ) (x-c) \\\\\n",
    "$$\n",
    "\n",
    "Taking limits gives and using the product rule for limits gives\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\lim_{x \\rightarrow c} (f(x)-f(c)) & = & \\left ( \\lim_{x \\rightarrow c}\\frac{f(x)-f(c)}{x-c} \\right ) (\\lim_{x \\rightarrow c} (x-c) ) \\\\\n",
    " & = & f'(c) \\cdot 0 .\n",
    "\\end{eqnarray}\n",
    "\n",
    "Thus, $\\lim_{x \\rightarrow c} = f(c)$.\n",
    "\n",
    "&square;\n",
    "\n",
    "**Note:** The converse is not true! Weierstrauss showed in 1872 that\n",
    "\n",
    "$$f(x) = \\sum_{n=0}^\\infty \\frac{1}{2^n} \\cos(3^n x)$$\n",
    "\n",
    "is continuous at every point, but does not have a derivative at any point (look up this result for the proof)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:52.067000Z",
     "start_time": "2019-10-09T22:14:51.897217Z"
    }
   },
   "outputs": [
    {
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/hEs7YOtXEPImnAqheedv8XbyZvGpxRLu+ZyEu3gk1TICYo1fOaqvGE1SQjhB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\nrkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0,1,100)\n",
    "f10 = sum([1/(2**n)*np.cos(3**n * x) for n in range(3)])\n",
    "f20 = sum([1/(2**n)*np.cos(3**n * x) for n in range(4)])\n",
    "f30 = sum([1/(2**n)*np.cos(3**n * x) for n in range(16)])\n",
    "plt.plot(x,f10,x,f20,x,f30);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T20:20:39.092606Z",
     "start_time": "2019-10-09T20:20:39.088602Z"
    }
   },
   "source": [
    "## Combinations of Derivatives\n",
    "\n",
    "**Thm 6:** If $f$ and $g$ are continuous at %c% then\n",
    "\n",
    "$$\n",
    "(af)'(x) = a f'(x)\n",
    "$$\n",
    "\n",
    "$$\n",
    "(f+g)'(x) = f'(x) + g'(x)\n",
    "$$\n",
    "\n",
    "$$\n",
    "(fg)'(x) = f'(x)g(x) + f(x)g'(x)\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\left (\\frac{f}{g} \\right)'(x) = \\frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}.\n",
    "$$\n",
    "\n",
    "**Example Proof of 3rd part:** Let $p = fg$. Then\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\frac{p(x) - p(c)}{x-c} & = & \\frac{f(x)g(x) - f(c)g(c)}{x-c} \\\\\n",
    "  & = & \\frac{f(x)g(x) - f(c)g(x) + f(c)g(x) - f(c)g(c)}{x-c} \\\\\n",
    "  & = & \\frac{f(x)-f(c)}{x-c} g(x) + f(c) \\frac{g(x)-g(c)}{x-c}\n",
    "\\end{eqnarray}\n",
    "\n",
    "Taking limits and noting that since $f$ and $g$ are continuous, the limits above equal $f(c)$ and $g(c)$ gives the desired result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:53.358905Z",
     "start_time": "2019-10-09T22:14:53.349004Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( 2 x e^{x} + \\left(x^{2} + 1\\right) e^{x}, \\quad 2 x e^{x} + \\left(x^{2} + 1\\right) e^{x}\\right )$$"
      ],
      "text/plain": [
       "⎛     x   ⎛ 2    ⎞  x       x   ⎛ 2    ⎞  x⎞\n",
       "⎝2⋅x⋅ℯ  + ⎝x  + 1⎠⋅ℯ , 2⋅x⋅ℯ  + ⎝x  + 1⎠⋅ℯ ⎠"
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# For example\n",
    "\n",
    "x = sm.symbols(\"x\")\n",
    "\n",
    "f = x**2 + 1\n",
    "g = sm.exp(x)\n",
    "\n",
    "(sm.diff(f,x)*g + f*sm.diff(g,x)), sm.diff(f*g,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the product rule extends to arbitrary length sums, so that\n",
    "\n",
    "$$\n",
    "(f_1 f_2 ... + f_n)'(x) = f_1'(x)f_2(x)...f_n(x) + f_n(x)f_2'(x)...f_n(x) + ... + f_n(x)f_2(x)...f_n'(x).\n",
    "$$\n",
    "\n",
    "Even better, if $f_i = f$ then,\n",
    "\n",
    "$$\n",
    "(f^n)'(x) = n f'(x)f(x)^{n-1}.\n",
    "$$\n",
    "\n",
    "And finally, if $f(x) = x^n$ then,\n",
    "\n",
    "$$\n",
    "\\frac{d}{dx} x^n = n x^{n-1}. \n",
    "$$\n",
    "\n",
    "Cool!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:54.633365Z",
     "start_time": "2019-10-09T22:14:54.628486Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$1000 x^{999}$$"
      ],
      "text/plain": [
       "      999\n",
       "1000⋅x   "
      ]
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.diff(x**1000,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The chain rule\n",
    "\n",
    "Since\n",
    "\n",
    "$$\n",
    "\\frac{g(f(x)) - g(f(c))}{x-c} = \n",
    "\\frac{g(f(x)) - g(f(c))}{f(x)-f(c)} \\cdot \\frac{f(x) - f(c)}{x-c}\n",
    "$$\n",
    "\n",
    "we get that\n",
    "\n",
    "$$\n",
    "(g \\circ f)'(x) = g'(f(x)) \\cdot f'(x).\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:56.869760Z",
     "start_time": "2019-10-09T22:14:56.857493Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left ( x^{2} + 1, \\quad e^{x}, \\quad 2 x, \\quad e^{x}, \\quad e^{x^{2} + 1}, \\quad 2 x e^{x^{2} + 1}, \\quad 2 x e^{x^{2} + 1}\\right )$$"
      ],
      "text/plain": [
       "⎛                       2            2            2    ⎞\n",
       "⎜ 2       x        x   x  + 1       x  + 1       x  + 1⎟\n",
       "⎝x  + 1, ℯ , 2⋅x, ℯ , ℯ      , 2⋅x⋅ℯ      , 2⋅x⋅ℯ      ⎠"
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# For example\n",
    "\n",
    "x = sm.symbols(\"x\")\n",
    "\n",
    "f = x**2 + 1\n",
    "g = sm.exp(x)\n",
    "gof = g.subs(x,f)\n",
    "\n",
    "df = sm.diff(f,x)\n",
    "dg = sm.diff(g,x)\n",
    "dgof = dg.subs(x,f)*df\n",
    "\n",
    "f, g, df, dg, gof, dgof, sm.diff(gof), "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## L'Hospital's Rule (0/0) case\n",
    "\n",
    "**Thm 8:** If $f$ and $g$ are differentiable at $a$ and $f(a) = g(a) = 0$ then\n",
    "\n",
    "$$\n",
    "\\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} = \\frac{f'(x)}{g'(x)}.\n",
    "$$\n",
    "\n",
    "**Proof:** Since $f(a) = g(a) = 0$,\n",
    "$$\n",
    "\\frac{f(x)}{g(x)} = \\frac{f(x) -f(a)}{g(x)-g(a)} = \\frac{\n",
    "\\frac{f(x) -f(a)}{x-a}\n",
    "}{\n",
    "\\frac{g(x) -g(a)}{x-a}\n",
    "} .\n",
    "$$\n",
    "&square;\n",
    "\n",
    "Note: You get to come up with examples in homework. \n",
    "\n",
    "Note: We have to have $f(a) = g(a) = 0$. For example, \n",
    "\n",
    "$$\n",
    "\\lim_{x \\rightarrow 0} \\frac{x+17}{2x+3} = \\frac{17}{3}\n",
    "$$\n",
    "\n",
    "but\n",
    "\n",
    "$$\n",
    "\\lim_{x \\rightarrow 0} \\frac{1}{2} = \\frac{1}{2} .\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Mean Value Theorem\n",
    "\n",
    "**Thm 7:** If $f : [a,b] \\rightarrow \\mathbb{reals}$ is continuous and has a derivative on $(a,b)$, then there is at least one point $c \\in (a,b)$ such that\n",
    "\n",
    "$$\n",
    "f(b) - f(a) = f'(c)(b-a) .\n",
    "$$\n",
    "\n",
    "We will not prove this here. See the reading.\n",
    "\n",
    "This theorem means that there is one point on the curve $y=f(x)$ at which the tangent line is parallel to the line from $(a,f(a))$ to $(b,f(b))$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-09T22:14:58.768167Z",
     "start_time": "2019-10-09T22:14:58.576805Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x120567d30>]"
      ]
     },
     "execution_count": 178,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Yc2YP5nwzawvXUtNSQ7h/ODcNvInMpEwGhLrneV0stwru3UUV3Pryl4R28+Gt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\nEMLN/D/0NJym8j/GZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(1, 2,100);\n",
    "f = 2*x + np.sin(7*x)\n",
    "fa = f[0]\n",
    "fb = f[-1]\n",
    "c = 1.6\n",
    "fc = 2*c + np.sin(7*c)\n",
    "plt.plot(\n",
    "    x,2*x + np.sin(7*x),\n",
    "    x,(fb-fa)/(2-1)*(x-2)+fb,\n",
    "    x,(fb-fa)/(2-1)*(x-c)+fc) \n",
    "plt.plot([c,c],[1,5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: Finding square roots\n",
    "\n",
    "**Example:** Find an approximation to $\\sqrt(105)$. \n",
    "\n",
    "Set $f(x) = \\sqrt{x}$. Then the MVT says that for some $c \\in (a,b)$\n",
    "\n",
    "$$\n",
    "f(b) - f(a) = \\frac{1}{2 \\sqrt{c}} (b-a)\n",
    "$$\n",
    "\n",
    "Choosing $a=100$ and $b=105$ we get that\n",
    "\n",
    "$$\n",
    "\\sqrt(105) - sqrt(100) = \\frac{5}{2 \\sqrt{c}} \n",
    "$$\n",
    "\n",
    "with $100 < c < 105$. Taking square roots gives $10 < \\sqrt{c} < \\sqrt{105}$. And since\n",
    "$\\sqrt{105} < \\sqrt{121} = 11$ we get \n",
    "\n",
    "$$\n",
    "10 < \\sqrt{c} < 11\n",
    "$$\n",
    "\n",
    "so that\n",
    "\n",
    "$$\n",
    "\\frac{5}{2\\cdot 11} < \\sqrt{105} - 10 < \\frac{5}{2 \\cdot 10}\n",
    "$$\n",
    "\n",
    "or $10.2272 < \\sqrt{105} < 10.2500$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: Inequalities\n",
    "\n",
    "**Example:** Show that $e^x \\geq 1+x$ for $x \\geq 0$. \n",
    "\n",
    "For $x = 0$ we get equality. So suppose $x>0$ and consider the interval [0,x]$. Then by the MVT\n",
    "\n",
    "$$\n",
    "e^x - e^0 = e^c (x-0)\n",
    "$$\n",
    "\n",
    "for some $c$ with $0 < c < 1$. Since $e^0 = 1$ and $e^c > 1$ we get the result."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Taylor's Theorem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Taylor's theorem can be regarded as an extension of the MVT for higher derivatives. It states:\n",
    "\n",
    "**Thm 9:** Say $f:I \\rightarrow \\mathbb{R}$is continuous and so are all of its derivatives. If $x_0 \\in I$, then \n",
    "\n",
    "$$\n",
    "f(x) = f(x_0) + f'(x_0)(x-x_0) + \\frac{f''(x_0)}{2!}(x-x_0)^2 + ... \n",
    "     + \\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n\n",
    "     + ... \n",
    "$$\n",
    "\n",
    "If an approximation is desired, then you can use\n",
    "\n",
    "$$\n",
    "f(x) = f(x_0) + f'(x_0)(x-x_0) + \\frac{f''(x_0)}{2!}(x-x_0)^2 + ... \n",
    "     + \\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n\n",
    "     + \\frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}\n",
    "$$\n",
    "\n",
    "for some $c$ with $x < c < x_0$. \n",
    "\n",
    "Taylor's theorem has many applications. We will return to it over and over. Some of the cooler observations are as follows.\n",
    "\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example:** $e^x$\n",
    "\n",
    "Note that all derivatives of $e^x$ are $e^x$. To find $e$ itself we can take $x_0 = 0$ and use\n",
    "\n",
    "$$\n",
    "e^x = e^0 + e^0 x + e^0 \\frac{x^2}{2!} + e^0 \\frac{x^3}{3!} + ...\n",
    "$$\n",
    "\n",
    "Simplifying, \n",
    "\n",
    "$$\n",
    "e^x = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + ...\n",
    "$$\n",
    "\n",
    "If you take this as a definition of $e^x$ then you can actually see why $\\frac{d}{dx}e^x = e^x$ since\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\frac{d}{dx}e^x & = & \\frac{d}{dx} \\left ( 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + ... \\right ) \\\\\n",
    "                & = & 0 + 1 + 2 \\frac{x}{2!} + 3 \\frac{x^2}{3!} + ... \\\\\n",
    "                & = & e^x. \n",
    "\\end{eqnarray}\n",
    "\n",
    "<hr>\n",
    "\n",
    "You can also make approximations to $e$ using the above.\n",
    "\n",
    "**Example:** Find $e$ with error less than $10^{-5}$. \n",
    "\n",
    "Taking $x=0$ and $x_0 = 1$ in Taylor's Theorem, the remainder is $\\frac{e^c}{(n+1)!}$ for some $c$ with $0 < c < 1$. Since we know that $e^c < 3$ we want $n$ so that \n",
    "\n",
    "$$\n",
    "\\frac{e^c}{(n+1)!} < \\frac{e^1}{(n+1)!} < \\frac{3}{(n+1)!} < 10^{-5}.\n",
    "$$\n",
    "\n",
    "or \n",
    "\n",
    "$$\n",
    "(n+1)! > 3 \\cdot 10^{5} = 300000.\n",
    "$$\n",
    "\n",
    "Since $9! = 362880$ but $8! = 40320$, we can take $n \\geq 9$. Then,\n",
    "\n",
    "$$\n",
    "e \\approx 1 + 1 + \\frac{1}{2!} + ... + \\frac{1}{8!} = 2.71827876984127 ...\n",
    "$$\n"
   ]
  },
  {
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   "execution_count": 179,
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    "ExecuteTime": {
     "end_time": "2019-10-09T22:15:02.439480Z",
     "start_time": "2019-10-09T22:15:02.434372Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2.71827876984127, 2.718281828459045, True)"
      ]
     },
     "execution_count": 179,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ep = sum([1/math.factorial(n) for n in range(9)])\n",
    "ep, math.e, abs(ep - math.e) < 10**(-5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conclusion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](https://3.bp.blogspot.com/-CAihtz6xvGw/WfYt5Wm4l7I/AAAAAAAAKIw/5siM_rslPfwIGP8I6XfrAatWz8ZvoxOqwCLcBGAs/s1600/i-thought-you-were-smart1.gif)"
   ]
  }
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