Irrationality exponents of certain fast converging series of rational numbers

Abstract

Let {xn} be a sequence of rational numbers greater than one such that xn+1 ≥ xn for all sufficiently large n and let εn ∈ {−1, 1}. Under certain growth conditions on the denominators of xn+1/x 2 n we prove that the irrationality exponent of the number ∑∞ n=1 εn/xn is equal to lim supn→∞(log xn+1/ log xn). 2010 Mathematics Subject Classification: 11A55, 11J70, 11J82, 11J91