Notation: $S_n = X_1 + \ldots + X_n$
Definition $1$ : The sequence of random variables $X_1, X_2, \ldots$ satisfies the Weak Law of Large Numbers if $$\frac{S_n - E[S_n]}{n} \overset{p}{\to} 0$$
Definition $2$: The sequence of random variables $X_{1}, X_{2}, \ldots$ satisfies the Strong Law of Large Numbers if $$\frac{S_n - E[S_n]}{n} \overset{\text{a.s.}}{\to} 0$$
Exercise: Let $(X_{n})$ be a sequence of independent random variables with $P(X_n = n^2) = \frac{1}{n^3}$, $P(X_n = 0) = 1 - \frac{1}{n^3}$. Does this sequence satisfy the Weak Law of Large Numbers? Does it satisfy the Strong Law of Large Numbers?
I've tried to find the distribution for $S_n$, but this approach didn't work. Any suggestions/hints on how to do this exercise are really appreciated.