# What is the sum of the infinite geometric series 1 + 1/5 + 1/25 +... ?

$1 + \frac{1}{5} + \frac{1}{25} + \cdots = \frac{5}{4}$
For the infinite geometric series $S = a + a q + a {q}^{2} + a {q}^{3} + \cdots$ we can use the formula $S = \frac{a}{1 - q}$ as long as $| q | < 1$.
In our case $a = 1 , q = \frac{1}{5}$ and $| q | = | \frac{1}{5} | = \frac{1}{5} < 1$ so
$S = \frac{1}{1 - \frac{1}{5}} = \frac{1}{\frac{4}{5}} = \frac{5}{4}$.