Series where the usual convergence tests fail Is there any simple series where all the usual convergence tests are inconclusive? And how is convergence/divergence determined for these series?
 A: In a spirit similar to Henning's answer: let $p_n$ denote the $n$th prime number, and set $a_n = 1$ if $p_n$ and $p_n + 2$ are both prime, and $0$ otherwise. Divergence of the series is of course equivalent to infinitude of twin primes. In fact, any statement about infinitude of subsets of countable sets can be mapped to an equivalent statement about convergence or divergence of infinite series.
Now, after massaging the statement above and recalling Godel's incompleteness theorem, one could see that in fact there are infinitely many infinite series whose convergence or divergence could not be determined given our (finite) axiomatic system.
A: How about
$$a_n = \begin{cases} 0 & \text{if $2n$ is the sum of two primes}\\ 1 & \text{otherwise}\end{cases} $$
A nontrivial amount of fame awaits you if you can determine whether that converges or not.
A: There are a lot of series for which we do not even know whether they converge or not. For instance, the convergence of the series $$\sum_{n} \dfrac{\mu(n)}{n^s}$$ for $\text{Re}(s) \in (1/2,1)$ is essentially equivalent to the Riemann Hypothesis. Hence, new techniques and ideas are needed to analyze convergence in this case.
There are plenty of other examples as well where we do not know about the convergence or divergence of the series. A trivial example is as follows. Any conjecture on infinitude of a certain thing can be converted into a series. For instance, if there is a conjecture stating there are infinitely many $x \in X$ such that a certain statement $P(x)$ is true. Then this can be converted into a series $$\sum_{x \in X} \mathbb{1}_{P(x)\text{ is true}}$$ and analyzing the convergence/divergence of this is equivalent to the conjecture being true or false.
A: For series with positive terms, Kummer's test always determines convergence. You will find this less exciting after seeing what the test is. 
A: It depends on what you think of as a "simple" series. There are many series where the convergence/divergence is unknown and extremely difficult. Here is a very relevant link. There are examples of series whose convergence is equivalent to the Riemann hypothesis and other famous conjectures. 
A: One more example is a Flint Hills series
$$
\sum_{k = 1}^{\infty} \frac{1}{n^3 \sin^2 n}.
$$
At the moment nobody knows whether it converges or not, because $\frac{1}{\sin^2 n}$ can have a sporadic large values.
The question of convergence of this series is closely related to the irrationality measure of $\pi$. In particular, the convergence would imply
$$
\mu (\pi) \le 2.5,
$$
while currently known best upper bound for the irrationlity measure is
$$
\mu(\pi) \le 7.6063...
$$
The details on the connection between the two can be found in the following paper of Alekseyev.
