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.

$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function.  This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$.
Its convergence is unknown if $1/2< s< 1$ (convergence in this interval is essentially the Riemann hypothesis).