Let $(X_{j})_{j\geq1}$ be independent and uniformly distributed on $(-j,j)$ and let $S_{n} = X_{1} + ... + X_{n}$. Show that $\lim_{n \to \infty}S_{n}/n^{3/2}=Z$ in distribution where $Z$ ~ $N(0,1/9)$ and that $\lim_{n \to \infty} \frac{S_{n}}{\sqrt{\sum_{j=1}^{n} \sigma_{X_{j} } }} = Y$ in distribution where $Y$ ~ $N(0,1)$.
For the former part, I am given a hint that we should use characteristic functions to show that the limit is $e^{-u^2/18}$ by using $\sum_{j=1}^{n}j^2 = n(n+1)(2n+1)/6$, but I don't understand how this helps at all because I can't seem to get a summation anywhere in my answer? I have shown that $\phi_{{X}_{j}}(u)= \frac{sin(uj)}{uj}$ and $\phi_{{S}_{n}}(u) = \prod_{j=1}^{n}\frac{sin(uj)}{uj}$, and I haven't been able to find a way to use this.
For the latter part, I showed that the denominator is $\sqrt\frac{n(n+1)(2n+1)}{18}$, and I don't know what to do with this either. I would appreciate any sort of guidance if possible.