• Corpus ID: 115161189

Summing the curious series of Kempner and Irwin

  title={Summing the curious series of Kempner and Irwin},
  author={Robert Baillie},
  journal={arXiv: Classical Analysis and ODEs},
  • 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 

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