PI estimation with Basel summation.ipynb

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   "source": [
    "# The Basel Problem for PI estimation"
   ]
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   "source": [
    "The Basel problem asks for the sum of reciprocals of squares of natural numbers\n",
    "<br><div style=\"padding-top:10px; padding-bottom:10px; text-align:center\"><span style=\"font-size:2em;\"> $\\sum_{1}^{\\infty}\\frac{1}{n^2}=1+\\frac{1}{2^2}+\\frac{1}{3^2}+...$</span></div>\n",
    "It has proven that this summation is <span style=\"font-size:2em\">$\\frac{\\pi^2}{6}$</span>\n",
    "<br>Instagram: @ostad.ai\n",
    "<br>GitHub: https://github.com/ostad-ai/PI-estimation-methods\n",
    "<br>www.linkedin.com/in/dr-hamed-shah-hosseini"
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   "source": [
    "First, we import the necessary libraries"
   ]
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   "source": [
    "import numpy as np"
   ]
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   "source": [
    "Then, we implement the Basel summation with $n$=nFinal"
   ]
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   "source": [
    "def Basel(nFinal=10000000):\n",
    "    ns=np.arange(1,nFinal+.01,1)\n",
    "    return np.sum(1./ns**2)"
   ]
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   "cell_type": "markdown",
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   "source": [
    " Now, we estimate PI by using the Basel summation as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The PI is: 3.1415925581\n"
     ]
    }
   ],
   "source": [
    "print('The PI is:',np.sqrt(6*Basel()))"
   ]
  }
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