On the rationality of Cantor and Ahmes series

We give criteria for the rationality of Cantor series ∑ ∞ n=1 bn a1···an and series ∑ ∞ n=1 bn an where a1, a2, · · · and b1, b2, · · · are integers such that an > 0 and the series converge. We precisely say when ∑ ∞ n=1 bn a1···an is rational (i) if {an} ∞ n=1 is a monotonic sequence of integers and bn+1 − bn = o(an+1) or lim supn→∞( bn+1 an+1 − bn an ) ≤ 0, and (ii) if bn+1 an+1−1 ≤ bn an−1 for all large n. We give similar criteria for the rationality of Ahmes series