Question

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

Answer 1

`5/4`

Explanation:

You are looking for

`S = \sum_{i=0}^\infty \frac{1}{5^n}= \sum_{i=0}^\infty (\frac{1}{5})^n`

In general,

`S = \sum_{i=0}^\infty a^n`

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

`S = \sum_{i=0}^\infty a^n = \frac{1}{1-a}`

So, in your case,

`S = \sum_{i=0}^\infty (\frac{1}{5})^n = \frac{1}{1-1/5} = \frac{1}{4/5} = 5/4`