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 $$ \sum _{n=1}^{\infty} \frac{1}{(n+1)(n+2)\cdots (n+f(n))}$$ has 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 Sōichi Kakeya on the set of all subsums of a convergent positive series. We also discuss why the same problem is likely difficult when $(f(n))_{n=1}^{\infty }$ is additionally assumed to be increasing.