Series whose convergence is not known

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

For an interesting example, take $\sum_{n=1}^\infty \frac{|\sin(n)|^n}{n}$. Deciding whether or not this converges seems to require more knowledge than is currently available about the rational approximations of $\pi$. The series $\sum_{n=1}^\infty \frac{|\sin(n t \pi)|^n}{n}$ converges for almost every real $t$ (in the sense of Lebesgue measure), but diverges for $t$ in a dense $G_\delta$ subset of $\mathbb R$.

EDIT: ... and now it is known to converge, as Sam Hopkins commented.

First, here's a silly example: define $a_n$ to be 1 if $n$ and $n+2$ are both prime. I think it's fair to say that that doesn't count as the kind of series that crops up in an elementary analysis course. But the real reason it's silly is that it is just an encoding of a problem that isn't about the convergence of a series at all.

Somehow I feel that the examples involving $\sin n$ are of a similar flavour, even though they resemble elementary series in analysis much more closely. When you try to solve them, you quickly find that the problem turns into something else (in this case, questions about rational approximations to $\pi$).

It's probably asking for too much, but I wonder whether there is a series whose convergence is not known, where one wouldn't get the feeling that the convergence of the series was just an artificial way of asking a different problem. The reason it may be asking too much is that any attempt to prove convergence is likely to involve a certain amount of reformulation, and who is to say what counts as changing the problem to a different one?

But let me attempt to ask a narrower question that gets somewhere in the right direction. Is there a series $(a_n)$ that has a relatively simple analytic definition (so you can't just encode some counting problem), such that every $a_n$ is non-negative, the $a_n$ are decreasing, the difference sequence $a_{n+1}-a_n$ is decreasing, the difference sequence of that is decreasing, and so on, and the convergence of the sum $\sum a_n$ is unknown?