Not a rigorous proof (I'll leave that to further investigation):

S(n) = 1/n + 1/n^2 + ...

= 1/n ( 1 + 1/n + 1/n^2 + ...) <--needs more justification in a rigorous proof

S(n) = 1/n ( 1 + S(n) )

Simple algebra from here:

n * S(n) - S(n) = 1

S(n) = 1 / (n-1)