Series whose convergence is not known 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.
 A: $1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ 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).
A: 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. 
A: This is a problem about convergence of a certain type of measures,
I am somewhat familiar with, which is yet to be proved:
Let $$ T = \sum_{j=1}^k Q_j \frac{d^j}{dx} $$
with $\deg Q_j\leq j$ with equality for at least one $j,$ but $\deg Q_k < k.$
There are unique monic eigenpolynomials $p_n$ of each degree $n>n_0,$ 
that is $Tp_n = \lambda_n p_n.$
Let $d := max_{j \in [j_0,k]} \frac{j-j_0}{j - \deg Q_j}$
where $j_0$ is the largest $j$ such that $\deg Q_j = j,$
and consider the polynomials $q_n(z)=p_n(n^dz).$
Conjecture: the root measures $\mu_n$ obtained by placing a point mas of weight $1/n$ at each root of $q_n$ converges weakly to a probability measure $\mu$ with has 
a compact support in the shape of a tree.
Reference: http://su.diva-portal.org/smash/get/diva2:196825/FULLTEXT01
A: 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?
A: 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).
A: Let $B_{k}$ refer to the Bernoulli numbers. Then consider the series
$$\sum_{n=0}^{\infty}\left(n+\frac{1}{2}\right)\biggr|\sum_{k=0}^{\infty}\frac{c_{2n+1,2k+1}}{2k+2}\log\left(\frac{2k+1}{2k+2}\frac{(-1)^{k}B_{2k+2}(2\pi)^{2k+2}}{2(2k+2)!}\right)\biggr|^{2}$$
where 
$$c_{2n+1,2k+1}=\frac{(-1)^{n-k}(2n+2k+2)!}{2^{2n+1}(n-k)!(n+k+1)!(2k+1)!}.$$
It is not known whether this series converges.  In fact, its convergence is equivalent to the Riemann Hypothesis.
(There are a lot of ways to make series that converge if and only if RH)
Hope that helps,
Reference: 
(1) John Corning Carey's Paper:  http://jcarey.best.vwh.net/RHHardy.pdf
A: You probably didn't intend this, but there are plenty of artificial ways to write down series whose convergence is equivalent to various types of unsolved problems. For example, let $S$ be any set such that it is an unsolved problem whether $S$ is infinite or not, and consider the convergence of the series $\sum_{s \in S} 1$. 
A: Assume that $A\subset\mathbb{N}$ contains no 3-term arithmetic progression. It was conjectured by Erdős and Turán that $\sum_{a\in A}\frac{1}{a}$ converges. As far as I know this is open, although we are getting closer to a proof, see Sanders' paper.
