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- W1861631309 abstract "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. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.007x10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol." @default.
- W1861631309 created "2016-06-24" @default.
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- W1861631309 date "2008-06-27" @default.
- W1861631309 modified "2023-09-27" @default.
- W1861631309 title "Summing the curious series of Kempner and Irwin" @default.
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