Does the following sequence of independent random variables with symmetric probability:
$\mathbb{P}(X_n = 1) = \frac{1}{2} - \frac{1}{2n^2}$ and $\mathbb{P}(X_n = n) = \frac{1}{2n^2}$
fulfils the Central Limit Theorem?
Now, I wish I could show my attempt but there isn't much to show. Should I first see if $\mathbb{E}X^2 < \infty$ and then check the Lindenberg's condition?