Finding the limit is actually the easy part. If you write a_n as
Σ (k/n)2 cos(kπ/n) (1/n),
you can recognize this as a Riemann sum with an equally spaced partition of the interval [0,1]. Note that for
Δx = 1/n, x_k = k/n, and f(x) = x2 cos(πx),
in the limit as n->∞, the sum becomes
∫ x2 cos(πx) dx = -2/π2.
I'll have to try to work on the convergence argument---if one separate from actually finding the limit is necessary.
I hope this at least helps!