Non convergence of a series of random variables

Question

Question:

Let $(X_n), n\in\mathbb{N}$ be a sequence of independent r.v.s such that $P(X_n=n^4)=\frac{1}{n^4}$ and $P(X_n=-1)=1-\frac{1}{n^4}$. Study the a.s. convergence of $S_n=\sum_{i=1}^n X_n$ as $n\rightarrow +\infty$.

My Attempt:

I have been simulating the stochastic process $S_n$ in R to understand whether convergence was possible at all but in none of the simulations I performed I obtained a finite value (all of the paths go to -9999).

Also, clearly, $\sum_{i=1}^n \operatorname{Var}(S_n)\rightarrow +\infty$ as $n\rightarrow +\infty$. Can I thus conclude that $S_n$ does NOT converge a.s.?

Many thanks in advance for the help!

Answer 1

Note that $P(X_n>1\quad \text{i.o})=0$ by Borel cantelli. Hence $X_n=-1$ for all but finitely many $n$ with probability $1$. In particular, the series diverges.