Irrationality exponents of certain fast converging series of rational numbers

Abstract

Let $\{x_n\}$ be a sequence of rational numbers greater than one such that $x_{n+1} \geq x^2_n$ for all sufficiently large $n$ and let $\varepsilon_n \in \{-1,1\}$. Under certain growth conditions on the denominators of $x_{n+1}/x^2_n$ we prove that the irrationality exponent of the number $\sum^{\infty}_{n=1} \varepsilon_n/x_n$ is equal to $\limsup_{n\to\infty}(\log x_{n+1}/\log x_n)$.