Another way could be this:
(1/5)+(2/5^2)+(3/5^3)+...
=[(1/5)+(1/5^2)+(1/5^3)+...]+(1/5)[(1/5)+(1/5^2)+(1/5^3)+...]+(1/5^2)[(1/5)+(1/5^2)+(1/5^3)+...]+...
=[(1/5)+(1/5^2)+(1/5^3)+...][1+(1/5)+(1/5^2)+(1/5^3)+...]
=[(1/5)/[1-(1/5)]][1+(1/4)]
=(5/4)(1/4)=5/16
Damn, I wish I knew latex..