TITLE: Testing for convergence $\sum_{j=1}^{\infty}\frac{1}{\sum_{i=1}^{j}p_i}$
QUESTION [7 upvotes]: How would we test for convergence the series below?
$$\sum_{j=1}^{\infty}\frac{1}{\sum_{i=1}^{j}p_i}$$
where $p_i$ is the $i$th prime number. I'd be glad to learn an elementary way. Thanks.

REPLY [7 votes]: by priem number theorem $$ \sum_{j=1}^{n}p_{j} \sim \frac{n^{2}ln(n)}{2} $$
so your series goes about $$ \sum_{n=2}^{\infty} \frac{2}{n^{2}ln(n)} $$
which is itself convergent..