# 0.999.../The geometric series formula

$\sum _{n=0}^{\infty }x^{n}=1+x+x^{2}+x^{3}+...$ is known as a geometric series.

If, $|x|<1$ , then this series converges to:

$\sum _{n=0}^{\infty }x^{n}={\frac {1}{1-x}}$ Proof: Define the partial sum $S_{n}$ :

$S_{n}=\sum _{n=0}^{n}=1+x+x^{2}+x^{3}+...x^{n-1}$ $xS_{n}=\sum _{n=0}^{n}=..+x+x^{2}+x^{3}+...x^{n-1}+x^{n}$ Note that both partial sums have n terms. When they are subtracted only the first term in and the last term in $xS_{n}$ will remain:

$S_{n}-xS_{n}=1-x^{n}$ , which can be easily solved for $S_{n}$ Note that we have failed to establish when the infinite series converges. This requires an understanding of the what happens when we take the limit of the partial sum as n goes to infinity. This is left to the reader as problem 1.

Sample problems:

1. Prove that sequence of partial sums, $\{S_{n}\}_{n=1}^{\infty }=\{S_{1},S_{2},S_{3},...\}$ , converges if $|x|<1$ .
2. This can be used to convert other repeating decimals into fractions. For pedagogical purposes, it is better to begin with a known fraction. From such a list, we consider: 0.181818... . Convert this into a rational fraction.

## Footnotes

1. It is your duty to improve Wikibooks. Please write up your solutions and post them here ASAP.
2. Note the two ways to write this sequence, the latter resembles set-theory notation, except that the order in which elements appear is important. See wikiversity:Set theory#Basic concepts and notation
3. See for example http://www.factmonster.com/ipka/A0876707.html