Summing a Curious, Slowly Convergent Series

diverges. Suppose we delete from the series all terms whose denominators, in base 10, contain the digit 9. Kempner [13] proved in 1914 that the remaining series converges. We also get convergent series by deleting terms whose denominators contain any digit or string of digits, such as "42", or "314159". However, these series converge so slowly that calculating their sums directly is out of the question. Here, we describe an algorithm to compute these and related sums to high precision. For example, the sum of the series whose denominators contain no "314159" is approximately 2302582.33386. We explain why this sum is so close to 106 log 10 by developing asymptotic estimates for sums that omit strings of length n, as ft approaches infinity. At first glance, it seems counter-intuitive that merely omitting the terms 1/9, 1/19, 1/29,... from the harmonic series would produce a convergent series. It appears that we are removing only every tenth term from the harmonic series. If that were the case, then the sum of the remaining terms would indeed diverge. This series converges because in the long run, we in fact delete almost everything from the harmonic series. We begin by deleting 1/9, 1/19, 1/29, .... But when we reach 1/89, we delete 11 terms in a row: 1/89, then 1/90 through 1/99. When we reach 1/889, we delete 111 terms in a row: 1/889, then 1/890 through 1/899, and finally 1/900 through 1/999. Moreover, the vast majority of integers of, say, 100 digits contain at least one "9" somewhere within them. Therefore, when we apply our thinning process to 100-digit denominators, we will delete most terms. Only 8 x 9"/(9 x 10") ^ 0.003% of terms with 100-digit denominators will survive our thinning process. Schumer [14] argues that the problem is that we tend to live among the set of puny integers and generally ignore the vast infinitude of larger ones. How trite and limiting our view! We can paraphrase Kempner's argument as follows. There are 8 x 9l~l integers with i digits that do not contain a "9". Their reciprocals are all at most 1/101"-1, so the sum of their reciprocals is at most 8 x (9/10)*-1. Summing these numbers over / gives a convergent geometric series that converges to 80. This is an upper bound of the sum of the reciprocals of integers not containing a "9". Kempner's reasoning and convergence result (but not his upper bound) apply to any digit in any base. That is, if J is a digit in base B, then if we delete from the harmonic series all terms that contain the base-Z? digit d, we likewise get a convergent series. We can use this fact to show that deleting terms that contain any fixed string of digits also gives a convergent series. Also, there is a connection between the set of numbers that contain the decimal string "42" and the set of numbers that, in base 100, have a digit equal to 42. The