Criterion for irrational sequences.(English)Zbl 0768.11021

The main result of the paper is the following. Let $$(a_ n)$$ and $$(b_ n)$$ be two sequences of positive integers such that $$a_ n \geq r^{2^ n}_ n$$ and $$b_{n+1} \leq r^ B_ n$$ hold for all sufficiently large $$n$$. Here $$(r_ n)$$ is a nondecreasing sequence of positive reals tending to infinity and $$B$$ is a positive integer. Then the sum of the series $$\sum_{n \geq 1}b_ n/a_ n$$ is an irrational number.