It is trivial that if there are finitely many twin primes then Brun’s constant must be a rational number. And GammaTester (below) has offered an example of an infinite series that converges to a rational number.
My question is, even if there are infinitely many twin primes, then how is it possible for Brun's constant to be (potentially) irrational, since it is an infinite always-positive sum of always-rational numbers? How can such a sum converge on an irrational number?