# How do I find the sum of the infinite geometric series 2/3, - 4/9, ...?

Oct 17, 2014

Recall: Geometric Series

$a + a r + a {r}^{2} + a {r}^{3} + \cdots = \frac{a}{1 - r}$ if $| r | < 1$.

Let us look at the posted geometric series,

$S = \frac{2}{3} - \frac{4}{9} + \frac{8}{27} - \cdots$

by rewriting a bit to fit the form of geometric series,

$= \frac{2}{3} + \frac{2}{3} \left(- \frac{2}{3}\right) + \frac{2}{3} {\left(- \frac{2}{3}\right)}^{2} + \cdots$

since $a = \frac{2}{3}$ and $r = - \frac{2}{3}$,

$= \frac{\frac{2}{3}}{1 - \left(- \frac{2}{3}\right)} = \frac{2}{5}$

I hope that this was helpful.