Infinite Sum
Posted: Tue Jun 21, 2011 8:38 pm
Let $a_1,a_2,\dots$ be a sequence of integers satisfying $0\le a_n<n$ for every $n$. Prove that the value of $\sum_{n=1}^\infty a_n/n!$ is irrational if and only if $a_n\le n-2$ for infinitely many $n$ and $a_m > 0$ for infinitely many $m$.
(From the Mathematical Tripos part IA, 2008. The original question is longer, but this is the interesting bit.)
(From the Mathematical Tripos part IA, 2008. The original question is longer, but this is the interesting bit.)