# How do you find the sum of the infinite geometric series 2+2/5+2/25+2/125+...?

##### 1 Answer
Nov 26, 2015

${\sum}_{i = 0}^{\infty} 2 \cdot {\left(\frac{1}{5}\right)}^{i} = 2.5$

#### Explanation:

General formula for an infinite geometric series sum $a + \left(a \cdot r\right) + \left(a \cdot {r}^{2}\right) + \left(a \cdot {r}^{3}\right) + \ldots$ with $\left\mid r \right\mid < 1$ is
$\textcolor{w h i t e}{\text{XXX}} \frac{a}{1 - r}$

For the given series $a = 2$ and $r = \frac{1}{5}$
So the sum is
$\textcolor{w h i t e}{\text{XXX}} \frac{2}{1 - \frac{1}{5}} = \frac{2}{\left(\frac{4}{5}\right)} = \frac{2 \cdot 5}{4} = 2 \frac{1}{2}$