A curious result related to Kempner's series

TL;DR

This paper explores the convergence properties of specific reciprocal digit series, revealing that the sum of reciprocals of integers with exactly r nines approaches 10 log 10 as r increases.

Contribution

It establishes a new asymptotic result for the series of reciprocals of integers with exactly r nines, extending previous convergence results.

Findings

The series of reciprocals of integers with exactly r nines converges.

As r approaches infinity, the sum tends to 10 log 10.

The result generalizes known convergence properties of digit-restricted series.

Abstract

It is well known since A. J. Kempner's work that the series of the reciprocals of the positive integers whose the decimal representation does not contain any digit 9, is convergent. This result was extended by F. Irwin and others to deal with the series of the reciprocals of the positive integers whose the decimal representation contains only a limited quantity of each digit of a given nonempty set of digits. Actually, such series are known to be all convergent. Here, letting $S^{(r)}$ $(r \in \mathbb{N})$ denote the series of the reciprocal of the positive integers whose the decimal representation contains the digit 9 exactly $r$ times, the impressive obtained result is that $S^{(r)}$ tends to $10 \log{10}$ as $r$ tends to infinity!