• Corpus ID: 115161189

# Summing the curious series of Kempner and Irwin

```@article{Baillie2008SummingTC,
title={Summing the curious series of Kempner and Irwin},
author={Robert Baillie},
journal={arXiv: Classical Analysis and ODEs},
year={2008}
}```
• Robert Baillie
• Published 27 June 2008
• Mathematics
• arXiv: Classical Analysis and ODEs
In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum is about 22.92068. In 1916, Irwin proved that the sum of 1/n where n has at most a finite number of 9's is also a convergent series. We show how to compute sums of Irwins' series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39โฆย

## Figures and Tables from this paper

In 1914 Kempner [5] showed that the series of all reciprocals of natural numbers without the digit 9 in their decimal expansion converges. This series turned out to be extremely slowly convergent toโฆ
In 1914, Kempner proved that the series consisting of the inverses of natural numbers which are free of the digit 9 is convergent. In 1916, Irwin considered the convergence problem of the seriesโฆ

## References

SHOWING 1-10 OF 19 REFERENCES

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 wasโฆ
• Mathematics
• 1974
We conjectured that the series (1.2) are irrational under the single assumption that {a,) is monotonic and we observed that some such condition is needed in view of the possible choices a, = cp(n) +โฆ
• E. T.
• Mathematics
Nature
• 1946
THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.โฆ
• P. Borwein
• Mathematics, Philosophy
Mathematical Proceedings of the Cambridge Philosophical Society
• 1992
Abstract We prove that the series are irrational and not Liouville whenever q is an integer (q โช 0, ยฑ1) and r is a nonzero rational (r โช โqn).
This book poses certain problems to a reviewer for the Gazette such as the fact that it is an expository account and the readers of the Gazette are professionalsโ but thatโs OK, as most of them, evenโฆ
This dictionary of numbers, arranged in order of magnitude, exposes the fascinating facts about certain numbers and number sequences. The aim of the book is to entertain and enthral the reader, whichโฆ
• Mathematics
Am. Math. Mon.
• 2008
An algorithm is described to compute convergent series by deleting terms whose denominators contain any digit or string of digits, such as "42", or "314159", and an upper bound of the sum of the reciprocals of integers not containing a "9" is found.