Summing the curious series of Kempner and Irwin

Details Summing the curious series of Kempner and Irwin Summing the curious series of Kempner and Irwin

2008-06-27

arxiv.org/abs/0806.4410v2

Mathematics

Classical Analysis and ODEs

27 pages

Scientific paper

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 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428708074784831968. The sum of 1/n where n has 100 zeros begins with the miniscule term 1/googol, yet its sum is about 10 ln(10) + 1.00745721706770421142x10^-197. Note that both of these sums are larger than that of Kempner's "no 9" series. We also show how to construct nontrivial subseries of the harmonic series that have arbitrarily large, but computable, sums. For example, the sum of 1/n where n has at most 434 occurrences of the digit 0 is about 10016.32364577640186109739.

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