Shoutbox

1 / n(n+1) = 1/n - 1/(n+1) for any n >=1.

Replacing all terms we get:
1/1 - 1/2 + 1/2 - 1/3 + ... + 1/n - 1/(n+1)

All terms cancel out in pairs, except first and last:
1/1 - 1/(n+1)

Which is what we need.