I am trying to show the following for two sums of random variables: Let ${X_n}, n∈ \mathbb{N}$ be a sequence of independent random variables.
First, assume that $P(X_n = 1) = P(X_n = −1) = \frac{1}{2}(1-\frac{1}{n^2})\\$ and $P(X_n = \sqrt{n}) = P(X_n = −\sqrt{n}) = \frac{1}{2n^2}$
Prove that for the sums $S_n = X_1 + · · · + X_n$ the Central Limit Theorem is valid.
Next consider $P(X_n = 1) = P(X_n = −1) = \frac{1}{2}(1-\frac{1}{n})\\$ and $P(X_n = \sqrt{n}) = P(X_n = −\sqrt{n}) = \frac{1}{2n}$
Prove that in this case the Central Limit Theorem does not hold.
I'm not really sure how I should approach this. Further more what is the intuition that the CLT falls apart when we remove that power of n from the denominator.