On the irrationality of certain super-polynomially decaying series

We give a negative answer to a question by Paul Erdős and Ronald Graham on whether the series $$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{(n+1)(n+2)\cdots (n+f(n))}$$ has an irrational sum whenever $(f(n){)}_{n=1}^{\mathrm{\infty }}$ 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 $(f(n){)}_{n=1}^{\mathrm{\infty }}$ is additionally assumed to be increasing.