TITLE: Series whose convergence is not known
QUESTION [57 upvotes]: For most of the mathematical concepts I learn, it has more or less always been possible  to find (at least google and find) unsolved problems pertaining to that specific concept. Keeping a bag of unsolved problems on most topics I know has been to my benefit in that it reaffirms me that mathematics is a thriving subject. 
Coming to point, I am unable to find an elementary series of the kind we know on real analysis courses whose convergence is an unsolved  problem. Please, share if you have any. 
Thanks.

REPLY [25 votes]: Convergence of $\sum_{k=1}^\infty (-1)^k \frac{k}{p_k}$ is unknown, where $p_k$ is k-th prime (Guy 1994, p. 203; Erdős 1998; Finch 2003, according to Eric Weisstein's MathWorld).