Weak convergence and limiting distribution

Question

I have $X_{i} \sim \operatorname{Unif}\left(0,1\right)$ iid random variables and have to show that $$ \frac{4\sum_{i=1}^n iX_{i} - n^2}{n^{3/2}}$$ converges weakly and compute its limit. How can I do this?

I would start with looking at $\sum_{i=1}^n iX_{i}$. Is there any Lemma so that I can replace $X_{i}$ by its expectation? Or is this a wrong idea?

Answer 1

Up to centering (adding a term which goes to infinity) and normalizing this looks the central limit theorem. But the problem is that we don't have a sum of identically distributed random variables. However, we can use Lyapunov's theorem.