Fractal Subseries of the Harmonic Series

TL;DR

This paper investigates the convergence properties of specific harmonic subseries derived from integer sequences with non-uniform digit distributions, including irregular sequences with undefined digit frequencies.

Contribution

It introduces a detailed analysis of harmonic subseries based on digit distribution patterns, including irregular sequences, expanding understanding of convergence criteria.

Findings

Certain digit distribution patterns lead to convergence of harmonic subseries.

Irregular sequences can produce both convergent and divergent subseries.

Examples illustrate the impact of digit frequency irregularities on series behavior.

Abstract

We study the convergence of certain subseries of the harmonic series corresponding to increasing sequences of integers whose digits in a certain base are not uniformly distributed. We also discuss the case of irregular sequences, where the frequency distribution of some of the digits does not exist. Examples are given for irregular sequences where the corresponding harmonic subseries is convergent, or divergent, respectively.