$\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function. It is known to converge for $s\ge 1$ and diverge for $s\le 1/2$. Its convergence is unknown if $1/2<s<1$, and convergence for all $s>1/2$ is equivalent to the Riemann hypothesis.

$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ 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).