# 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} }

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 + 1/39โฆย

## 2 Citations

### On the convergence of thinned harmonic series

- Mathematics
- 2015

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 toโฆ

### On the series of Kempner-Irwin type

- Mathematics
- 2009

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 seriesโฆ

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