$$x_1,...,x_n \sim \text{uniform (0,1)}$$
$$Y_n=\sum_i^n X_i$$
I want to find limit distribution by using central limit theorem.
$E(Y_n)=n/2$ and $V(Y_n)=n/12$
And Moment generating function $M_{Yn }(t)= (\frac{e^t-1}{t})^n$
What I did is the following. (Sorry for photo, but It is too hard to write such a long solution with latex for me)
I stack here. What is the distribution of Yn? If my solution is false, ignore it and solve correctly please