Erdős Problem #270

Let \(f(n)\to \infty\) as \(n\to \infty\). Is it true that \[\sum_{n\geq 1} \frac{1}{(n+1)\cdots (n+f(n))}\] is irrational? Erdős and Graham [ErGr80] write 'the answer is almost surely in the affirmative if \(f(n)\) is assumed to be nondecreasing'. The case \(f(n)=n\) was subsequently settled by Crmarić and Kovač [Ko26]. Hansen [Ha75] has shown that \[\sum_n \frac{1}{\binom{2n}{n}}=\sum_n \frac{n!}{(n+1)\cdots (n+n)}=\frac{1}{3}+\frac{2\pi}{3^{5/2}}\] is transcendental. Crmarić and Kovač [CrKo25] have shown that the answer to this question is no in a strong sense: for any \(\alpha \in (0,\infty)\) there exists a function \(f:\mathbb{N}\to\mathbb{N}\) such that \(f(n)\to \infty\) as \(n\to\infty\) and \[\sum_{n\geq 1} \frac{1}{(n+1)\cdots (n+f(n))}=\alpha.\]