Does the series $\sum_{i=1}^{\infty} \frac{X_i}{3^i}$ converge almost surely?

Question

Let $X_1,X_2,...$ be identically distributed random variables and $P(X_1=0)= \frac12=P(X_1 =2)$.

Does the series $\sum_{i=1}^{\infty} \frac{X_i}{3^i}$ converge almost surely?

I am getting an intuition of using Borel Canteli Lemma, but could not proceed with the problem. Need some Hints.

Answer 1

Hint: use the fact that $0\leq \frac{X_i}{3^i}\leq \frac{2}{3^i}$ for all $i$ to compare your series to a geometric series.