you have a geometric series
since you are summing the powers of 1/5, this converges
(1/5 < 1)
it's equal to 1/(1-1/5)
=1/(4/5) = 5/4
for geometric series, the sum is always
x = 1/(1-r)
where r is the ratio of successive terms.
if you set the sum equal to x:
x = 1+r+r^2+r^3...
and multiply each term by r
rx= r+r^2+r^3+....
then subtract
x-rx = 1+r-r+r^2-r^2+r^3-r^3+....
x-rx = 1
x(1-r)=1
x=1/(1-r)
