Question

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

Answer 1

`1+1/5+1/25+cdots=5/4`

For the infinite geometric series `S=a+aq+aq^2+aq^3+cdots` we can use the formula `S=a/(1-q)` as long as `|q|<1`.
In our case `a=1, q=1/5` and `|q|=|1/5|=1/5<1` so
`S=1/(1-1/5)=1/(4/5)=5/4`.