On the irrationality of certain super-polynomially decaying series

Abstract

We give a negative answer to a question by Paul Erd\H{o}s and Ronald Graham on whether the series \[ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)\cdots(n+f(n))} \] has an irrational sum whenever $(f(n))_{n=1}^{\infty}$ is a sequence of positive integers converging to infinity. To achieve this, we generalize a classical observation of S\={o}ichi Kakeya on the set of all subsums of a convergent positive series. We also discuss why the same problem is likely difficult when $(f(n))_{n=1}^{\infty}$ is additionally assumed to be increasing.