Solution
= Solution
Uniformly for the arguments involved,
$$
\log\frac{\sin x}{x}=-\frac{x^2}{6}+O(x^4).
$$
Hence
$$
\log\phi_{n^{-3/2}S_n}(\xi)
=-\frac{\xi^2}{6n^3}\sum_{j=1}^nj^2
+O\left(\frac{\xi^4}{n^6}\sum_{j=1}^nj^4\right)
\longrightarrow-\frac{\xi^2}{18},
$$
because $n^{-3}\sum j^2\to1/3$ and the error is $O(n^{-1})$. The limiting characteristic <function> is that of $N(0,1/9)$. Lévy's theorem gives
$$
\boxed{n^{-3/2}S_n\Rightarrow N(0,1/9).}
$$