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
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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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