This is the problem, I have done for case $p = 1/2$. But I have no clue where case $p \neq 1/2.$ Here is my attempt.
\begin{align*} \sum_{n=1}^{\infty} P(A_n) &= \sum_{n=1}^{\infty} P(S_{2n} = 0)\\ &= \sum_{n=1}^{\infty} \frac{(4pq)^n}{\sqrt{n \pi}}. \end{align*}
Do I need to compare this to the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n\pi}}$?, btw I don't know why I'd get the series above is less than $\infty.$
Thanks in advance.