Curious convergent series of integers with missing digits

TL;DR

This paper explores the convergence properties of series formed by reciprocals of integers with missing digits, extending Kempner's classical theorem to broader sets of integers.

Contribution

It generalizes Kempner's theorem to larger families of missing digit sets, analyzing their impact on the convergence or divergence of harmonic series.

Findings

Certain missing digit sets lead to convergent series

Other missing digit sets result in divergent series

The paper characterizes conditions for convergence and divergence

Abstract

A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with both convergent and divergent harmonic series.