Summing the curious series of Kempner and Irwin

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, among other things, 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 23.04428 70807 47848 31968. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.00745 x 10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol. Finally, we discuss a class of related series whose summation algorithm has not yet been developed.