TITLE: Convergent rational series: which ones remain rational?
QUESTION [8 upvotes]: Due to closure under addition, it is obvious that a finite sum of rationals is rational. The infinite ones, however (assuming they don't diverge), may remain rational, such as $\sum_{n \in \mathbb{N}} 2^{-n}$, or not, like $\sum_{n \in \mathbb{N}} n^{-2}$.
Is there a criterion to find out whether a (convergent) series of rationals is rational or irrational without calculating it?
P. S.: insights on the analogous question with infinite products are welcome.

REPLY [4 votes]: In fact one can reduce a very general class of problems to the question of determining whether a certain series of rational numbers converges to a rational number or not. Let $A$ be a subset of the natural numbers (for example, the set of twin primes). Then $A$ is infinite if and only if
$$\sum_{a \in A} 2^{-a^2}$$
is irrational, since if $A$ is infinite then the base-$2$ expansion of the above number cannot be eventually periodic. Many, many arbitrarily hard problems can be encoded as the problem of determining whether a set of natural numbers is infinite; for example, it is undecidable whether the set of natural numbers a Turing machine recognizes is infinite by Rice's theorem. 
In other words, what often doesn't come through in a calculus course is that the only series you've ever seen summed are likely to be much simpler than a "generic" series, which can be arbitrarily complicated. So one has to place strong restrictions on what kind of series are considered to have any hope of saying something general.