# How do I find the sum of the infinite series 1 + 1/5 + 1/25 +... ?

May 27, 2018

$\frac{5}{4}$

#### Explanation:

You are looking for

$S = \setminus {\sum}_{i = 0}^{\setminus} \infty \setminus \frac{1}{{5}^{n}} = \setminus {\sum}_{i = 0}^{\setminus} \infty {\left(\setminus \frac{1}{5}\right)}^{n}$

In general,

$S = \setminus {\sum}_{i = 0}^{\setminus} \infty {a}^{n}$

converges if and only if $| a | < 1$, which is obviously our case. If $| a | < 1$, the sum evaluates to

$S = \setminus {\sum}_{i = 0}^{\setminus} \infty {a}^{n} = \setminus \frac{1}{1 - a}$

$S = \setminus {\sum}_{i = 0}^{\setminus} \infty {\left(\setminus \frac{1}{5}\right)}^{n} = \setminus \frac{1}{1 - \frac{1}{5}} = \setminus \frac{1}{\frac{4}{5}} = \frac{5}{4}$