Finding an example where the sum of iid rv's is infinite almost surely $Y_1, Y_2, \dots$ are iid non-negative random variables. Let $S = \sum_{n=1}^\infty \alpha^n Y_n$, where $\alpha \in (0,1)$. Now, $EY < \infty$ implies that $S < \infty$ almost surely.
Can anyone give an example where $S=\infty$ almost surely by removing the $EY<\infty$ constraint? Thanks!

Assume that $Y_1$ is such that $\mathbb P\{|Y_1|>t\}\geqslant 1/\log t$ for $t$ large enough. Then for all $A$ the series $\sum_{n=1}^{+\infty}\mathbb P\{\alpha^n|Y_n|>A\}$ is divergent and using the Borel-Cantelli lemma we obtain that $\sum_{n=1}^{+\infty}\alpha^nY_n$ is almost surely divergent.
