An extension of a result of Sierpiński
The main results of this paper are the following theorems. Theorem 1: Let \(\{a_n\}^\infty_{n=1}\) and \(\{b_n\}^\infty_{n=1}\) are two sequences of positive integers such that \(\{b_n\}^\infty_{n=1}\) is bounded and for a fixed \(\lambda> 2\), \(\liminf_{n\to\infty} a_{n+1}/(a^{\lambda+1}_n)> 1\). Then the sum of the series \(\sum^\infty_{n=1}(- 1)^nb_n/a_n\) is a transcendental number. Theorem 2: Let \(\alpha\) and \(\beta\) are positive real numbers. Suppose that \(\{a_n\}^\infty_{n=1}\) and \(\{b_n\}^\infty_{n=1}\) are two sequences of positive integers such that for every sufficiently large \(n\), \(2^{(3+\alpha)^n}\leq a_n\) and \(b_n\leq 2^{(3+\beta)^n}\). Suppose that \(\{b_n/a_n\}^\infty_{n=1}\) is strictly monotone decreasing for \(n\) sufficiently large. Then the sum of the series \(\sum^\infty_{n=1}(- 1)^n b_n/a_n\) is a transcendental number. The proofs of these theorems are based on Roth's theorem.
scientific article; zbMATH DE number 2082494