Let $\{\eta_k\}_{k \in \mathbb{N}}$ be a family of i.i.d. random variables with parameter $p \in [0,1]$ Bernoulli distribution.
Let $S_n = \sum_{k = 1}^n \eta_k$. How to prove that $$\sum_{n \geq 1} \frac{\eta_n - p}{S_n + 1}$$ converge almost surely?
I tried to use the fact that $\frac{S_n}{n}$ goes to $p$ a.s. because we now that $\sum_{n \geq 1} \frac{\eta_n - p}{n}$ converges a.s., but i don't get success...
Thanks!