A criterion for linear independence of series (Q1880973)

Let \(\{a_{i,n}\}_{n=1}^\infty\) and \(\{b_{i,n}\}^\infty_{n=1}\) \((i= 1,\dots, s)\) be sequences of positive integers satisfying certain growth-order conditions (omitted here ). The author proves that the series \(\sum^\infty_{n=1} b_{i,n}/a_{i,n}\) \((i= 1,\dots, s)\) and the number 1 are linearly independent over \(\mathbb{Q}\). Using this criterion the irrationality of certain special series, which consist of rational numbers, are obtained. Moreover, some open problems are proposed.