I'm given a set of random variables $Z_n$ that have the following properties:
$\mathbb{P}\left [ Z_n = \frac{1}{n} \right ] = \frac{1}{n^2} $ and $\mathbb{P}\left [ Z_n = 0 \right ] =1- \frac{1}{n^2} $ I'm asked to find the value of: $\mathbb{P}\left [ \sum_{n=0}^\infty Z_n < \infty\right ] $
I've tried using the Borel–Cantelli lemma as the statement handles almost sure convergence but I didn't find a simple way to quantify the convergence of the series in way that would allow me to use the lemma. The only think that came to my mind is Cauchy's convergence test and it doesn't seem to be helping me much.
Note that $P(Z_n>0 \;\text{i.o})=0$ since $\sum_{n=1}^\infty n^{-2}<\infty$. Hence $Z_n=0$ eventually with probability one. It follows that $\sum_{n=0}^\infty Z_n<\infty$ with probability one.