On the irrationality of certain super-polynomially decaying series

We give a negative answer to the question by Paul Erdős and Ronald Graham on whether the series ∑n=1∞1(n+1)(n+2)⋯(n+f(n)) has irrational sum whenever (f(n))∞n=1 is a sequence of positive integers converging to infinity. To achieve this, we generalize a classical observation of Sōichi Kakeya on the set of all subsums of a convergent positive series. We also discuss why the same problem is likely difficult when (f(n))∞n=1 is additionally assumed to be increasing.