Added notebook about Geometric Series

11/Geometric_Series.ipynb

@@ -0,0 +1,73 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Geometric Series\n",
+    "In mathematics, a geometric series is a series with a constant ratio that is common between successive terms. \n",
+    "\n",
+    "### Definition\n",
+    "Any series which can be written in the following forms is a geometric series. The series is __generated by multiplying each term by a common ratio.__<br/>\n",
+    "\n",
+    "$$\n",
+    "\\sum_{n=1}^{\\infty} ar^{n-1} \n",
+    "\\quad\n",
+    "\\sum_{n=0}^{\\infty} ar^n\n",
+    "$$\n",
+    "\n",
+    "in which $a$ and $r$ are fixed real numbers, and $a \\ne 0$\n",
+    "\n",
+    "<small>__Note:__ These two series are identical, except for a _shift in the index variable_ from $1$ to $0$<br/>(not unlike the difference between a 0-indexed array and a 1-indexed array in computer science):</small>\n",
+    "\n",
+    "### Common Ratio\n",
+    "The common ratio is found by dividing any two consecutive terms:\n",
+    "$\\frac{a_n+1}{a_n}$\n",
+    "\n",
+    "### Examples\n",
+    "\n",
+    "$$\n",
+    "\\sum_{n=0}^{\\infty} {a_n} = \\left\\{ (1)(2)^n \\right\\} = 1 + 2 + 3 + 4 + ... + a_n\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\sum_{n=1}^{\\infty} {b_n} = \\left\\{ (1)\\left( \\frac{1}{2} \\right)^{n-1} \\right\\} = 1 + \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8} + ... + b_n\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\sum_{n=1}^{\\infty} {c_n} = \\left\\{ (-1)^{n+1}\\left(\\frac{1}{2}\\right)^{n} \\right\\} = 1 -\\frac{1}{2} + \\frac{1}{4} - \\frac{1}{8} + \\frac{1}{16} - \\frac{1}{32} + ... - c_n\n",
+    "$$\n"
+   ]
+  },
+  {
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+   "execution_count": null,
+   "metadata": {
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+   "outputs": [],
+   "source": []
+  }
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