Vtgyjfy

Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will eventually be shown to converge or diverge?

EDIT: People were kind enough to point out that ,without imposing restrictions on the terms, it's trivial to find such "open problem" sequences. So, to clarify, what I had in mind were sequences whose terms are composed of "simple" functions, the kind you would find in an introductory calculus text: exponential, factorial, etc.

It is unknown whether
$$
sum_{n=1}^inftyfrac{1}{n^3sin^2n}
$$
converges or not. The difficulty here is that convergence depends on the term $nsin n$ not being too small, which in turn depends on how well $pi$ can be approximated by rational numbers. It is possible that, if $pi$ can be approximated `too well' enough by rationals, then this will diverge. See this MathOverflow question for a discussion of this particular series.

Another even simpler example of a sequence (no summation) for which it is not known whether it converges or not is
$$
x_n=frac{1}{n^2sin n}.
$$
We would expect this to tend to zero, but the proof is beyond what is currently known. Suppose that there were only finitely many rational numbers $p/q$ with $vert p/q-pivertle q^{-3+epsilon}$ (for any $epsilon > 0$), then $x_n$ would tend to zero at rate $O(n^{-epsilon})$. If, on the other hand, there were infinitely many rationals satisfying $vert p/q-pivertle q^{-3-epsilon}$, then infinitely many $x_n$ would be of order at least $n^epsilon$, so it diverges. This can be expressed in terms of the irrationality measure of $pi$. The sequence $x_n$ converges to zero if the irrationality measure of $pi$ is less than 3, and diverges if it is geater than 3. Currently, the best known bound for the irrationality measure is that it is no more than about $7.6063$ (see the link to the mathworld page above). It is expected that the irrationality measure of $pi$ is 2 (it is known that all but a zero-measure set of real numbers have irrationality measure 2). Therefore, it is expected that $x_n$ tends to zero, but there is currently no proof of this.

As kind of a joke answer, but technically correct, and motivated by Chandru's deleted reply,
$$sum_{n=0}^infty sin(2pi n!,x)$$
where $x$ is the Euler-Mascheroni constant, or almost any other number whose rationality has not been settled. (If $x$ is rational, the series converges. The implication does not go the other way.)

It is unknown whether the series:
$$sum_n frac{(-1)^n n}{p_n}$$
converges. Here, $p_n$ is the $n$-th prime number. This problem is posed in Guy's book on unsolved problems in number theory and I am pretty sure that it originated from Erdős.

The Riemann hypothesis is that $sum_{n=2}^infty frac{Lambda(n)-1}{n^{1/2}log^{3+epsilon} n}$ converges for any $epsilon > 0$ (see here for a discussion of that $epsilon$).