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.
Categories: math.NT math.CA