Help me please to solve my homework.

Let X1,X2,...Xn be independent random variables such that P(Xk = k) = P(Xk = -k) = 1/(2k^2) and

P(Xk = 1) = P(Xk = -1) = 1/2(1-1/k^2).

Let Sn=∑Xk (k goes from 1 to n).

Prove, that (1/√n)Sn converges in distribution to N(0,1) and var((1/√n)Sn) converges to 2. (n goes to infinity) Thanks;o)

Julia