Let $(X_n)_{n\in \mathbb N}$ be independent continuous random variables with cdf $$f_n(x) := f_{X_n}(x) = \frac{n+1}{2}\lvert x \rvert ^n \mathbb 1_{[-1,1]}.$$ Let $S_n := \sum_{k=1}^nX_k$ and calculate the weak limit of $S_n/\sqrt{n}.$
Approach: The first thing I tried is to find the density of $X_1 + X_2$ by calculating the convolution of $f_1, f_2$ but that gets a little messy to calculate.
Also, trying to calculate the characteristic function of $X_n$ is not very smooth either (You could use partial integration $n$ times), there must be an easier way. Is there maybe a way to invoke the central limit theorem by modifying stuff? Any help appreciated!
It seems that Lyapunov's conditions applies. First observe that $X_k$ has expectation zero. The variance of $X_k$ is therefore the expectation of $X_k^2$, which is $1+\varepsilon_k$, where $\varepsilon_k\to 0$. Consequently, the sum of variances is order $n$. One can also compute the moments of order $2+\delta$ and find that $\mathbb E\left[\left\lvert X_k\right\rvert^{2+\delta}\right]=1+\varepsilon'_k$ where $\varepsilon'_k\to 0$.