A Class of Simple Rearrangements of the Alternating Harmonic Series

TL;DR

This paper introduces a family of involutive permutations of natural numbers that allow explicit rearrangements of the alternating harmonic series, with the digamma function as a key computational tool, and shows these rearrangements are dense in the real numbers.

Contribution

It defines a simple, explicit class of permutations for rearranging the alternating harmonic series and demonstrates their density in the real numbers.

Findings

Explicit formulas for rearrangements using the digamma function

The set of rearrangements is dense in the real numbers

Permutations are involutions, simplifying analysis

Abstract

We present an easily defined countable family of permutations of the natural numbers for which explicit rearrangements (i.e., the sums induced by the permutations) can be computed. The digamma function proves to be the key tool for the computations found here for the alternating harmonic series. The permutations $\phi$ under consideration are simple in a sense: they are involutions ($\phi\circ\phi$ is the identity function). We show that the countable set of rearrangements obtained from the simple involutions considered below are dense in the reals.