This is the last part of a three part problem on characteristic functions, and it's been driving me crazy over the last few days. Any help would be most appreciated.
$X_1,X_2, \ldots, X_n$ are independent, with $P(X_j=j)=P(X_j=-j)=1/2j, P(X_j=0)=1-1/j$ . Show that $S_n/n$ converges to a distribution with characteristic function $ \exp \left( -\int_0^1 \frac{1-\cos(xt)}{x} dx \right) $.
I have made some progress, but I end up with $\log \psi(t)=\sum_{j=1}^n \frac{\cos(tj/n)-1}{j}.$ This is close, but still wrong. Any thoughts?