On the rationality of Cantor and Ahmes series

AbstractWe give criteria for the rationality of Cantor series and series where a1, a2, ··· and b1, b2, ··· are integers such that an > 0 and the series converge. We precisely say when is rational (i) if {an}n=1∞ is a monotonic sequence of integers and bn + 1 − bn = o(an + 1) or lim and (ii) if for all large n. We give similar criteria for the rationality of Ahmes series and more generally series . For example, if bn > 0 and lim , where An = lcm(a1, a2, ···, an), then is rational if and only if for large n.On the other hand, we show that such results are impossible without growth restrictions. For example, we show that for any integers d > c > 1 there is a sequence {bn}n = 1∞ such that every number x from some interval can be represented as with an ∈ {c,d} for all n