Let $X_1, X_2, \dots$ be i.i.d. uniformly distributed random variables on $[0,1].$ I am trying to show that the sequence of random variables:
$$ \frac{4\sum_{i = 1}^n i X_i - n^2}{n^{3/2}} $$
converges weakly, and to find the distribution it converges to. I start out by trying to find the limiting distribution. Since I don't know how to handle adding uniform random variables of differing sizes, my only idea is to take the characteristic function of the $n$th term, which by rearranging the integrals is equal to
$$ e^{\frac{-itn^2}{n^{3/2}}} \int_0^1 e^{\frac{4x_1}{n^{3/2}} it} dx_1 \cdot \dots \cdot \int_0^1 e^{\frac{4nx_n}{n^{3/2}} it} dx_n.$$
By integrating we get that this is equal to
$$ e^{-\sqrt{n}it} \bigg( \frac{n^{3/2}}{4it} \bigg)^n \bigg( e^{\frac{4 \cdot 1}{n^{3/2}}it} - 1 \bigg) \dots \bigg( e^{\frac{4 \cdot n}{n^{3/2}}it} - 1 \bigg), $$ which I can't make any sense out of. Did I perform this evaluation correctly? If so, what distribution has a characteristic function resembling this? Otherwise, is there a better way to solve this than by using characteristic functions? I can write out more of my computation steps if they are needed.