On Riemann's Rearrangement Theorem for the Alternating Harmonic Series

Abstract

Let pn be the number of consecutive positive summands and let qn be the number of consecutive negative summands that appear in the classical Riemann’s rearrangement of the alternating harmonic series to sum a prescribed real number s. Assume that s > log2 and let x = (1=4)e 2s . It is shown that the sequence qn is constant equal to 1, and that the values of pn become stabilized: Eventually pn = bxc or pn =bxc + 1. Moreover, it is shown that x is rational if and only if the sequence pn is eventually periodic. The sequence pn is eventually constant if and only if x is integer, in which case pn =bxc for n big enough. Similar results are also true for s < log2.