An easy proof of the $\zeta(2)$ limit in the random assignment problem

The edges of the complete bipartite graph K-n,K-n are given independent exponentially distributed costs. Let C-n be the minimum total cost of a perfect matching. It was conjectured by M. Mezard and G. Parisi in 1985, and proved by D. Aldous in 2000, that C-n converges in probability to pi(2)/6. We give a short proof of this fact, consisting of a proof of the exact formula 1+1/4+1/9+...+1/n(2) for the expectation of C-n, and a O(1/n)bound on the variance.