## Infinite Sum

For college and university level advanced Mathematics
nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### Infinite Sum

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.)
"Everything should be made as simple as possible, but not simpler." - Albert Einstein