A theorem on irrationality of infinite series and applications

Here and in the sequel we maintain the convention of [Bad] that all series which appear are supposed to be convergent. Moreover, (an) and (bn), n ≥ 1, always denote sequences of positive integers. Also, for the sake of brevity, we simply say “the series ∑∞ n=1 bn/an is irrational” instead of “the sum of the series ∑∞ n=1 bn/an is an irrational number”. We note that Theorem A is, in a certain sense, best possible. Indeed, for a given sequence (bn) of positive integers and a given positive integer t, we define the sequence (wn), wn = wn(bn, t), by w1 = 1 + tb1 and wn+1 = 1 + tbn+1w1 . . . wn for n ≥ 1. By induction one easily checks that the following equalities are true: