Are there any series whose convergence is unknown? 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.

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 $\epsilon$).
