# A curious result related to Kempner's series

• Published in 2008
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!

## Other information

key
type
article
2023-02-27
date_published
2008-03-22

### BibTeX entry

@article{AcuriousresultrelatedtoKempnersseries,
type = {article},
title = {A curious result related to Kempner's series},
author = {Bakir Farhi},
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!},
comment = {},
}