Since this is everyone's favorite example of telescoping sums, let's do it another way just for giggles.
Combinatorial proof
The denominator is P(n+1, 2) which is the number of ways for 2 specified horses to finish 1st and second in an n+1 horse race.
So imagine you're racing against horses numbered {1, 2, 3, ....}.
Either you win, which has probability 0 in the limit, or there is a lowest numbered horse, n, that finishes ahead of you.
The probability that you beat horses {1,2, ... , n-1} but lose to n is (n-1)! / (n+1)! or P(n+1, 2) or 1/(n^2^+n), the nth term of the series.
Summing these mutually exclusive cases exhausts all outcomes except the infinitesimal possibility that you win.
Therefore the infinite sum is exactly 1.