I have a sequence of random variables $Y_i$ where $i\in\mathbb{N}$ are $uniform(0,1)$ random variables, define $X_k=kY_k$ also let $S_n=\sum_{k=1}^nX_k$
I want to use characteristic functions to show that the item on LHS converges weakly to the constant 1.
$$Z=\frac{S_n}{\frac{n^2}4}\rightarrow^{w}1$$
In essense I want to show that $\phi_Z(t)=\mathbb{E}(e^{itY})$ converges to $e^{it}=\phi_1(t)$
I know that the characteristic functions of the $X_k$ are given by $$\phi(t)=\frac{1}{itk}(e^{itk}-1)=1+\frac{itk}2+\frac{(itk)^2}6+...$$
and so $$\phi_Z(t)=(\phi(\frac{4t}{n^2}))^n=(1+\frac{2itk}{n^2}+..)^n$$
I am trying to get into a form where i can use the fact that $e^x=lim_{n\rightarrow\infty}(1+\frac{x}n)^n$
can somebody help me finish.
I do not know whether the approach with characteristic functions is the simplest. One can also argue directly, by showing that $$ ~\lim_{n\to +\infty}\mathbb E\left[\left(\frac 1n\sum_{k=1}^nkX_k-\mathbb E\left[kX_k\right]\right)^2\right]=0. $$