- Kempner series
The Kempner series is a modification of the harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains a 9 digit. That is, it is the sum:where the prime indicates that "n" takes only values whose decimal expansion has no 9s. The series was first studied by A. J. Kempner in 1914.cite journal
last = Kempner
first = A. J.
year = 1914
month = February
title = A Curious Convergent Series
journal = American Mathematical Monthly
volume = 21
issue = 2
pages = 48–50
publisher = Mathematical Association of America
location = Washington, DC
issn = 00029890 ] The series is interesting because of the counter-intuitive result that unlike the harmonic series it converges (Kempner showed this value was less than 80 and to 20 digits the actual sum is 22.92067 66192 64150 34816).Schmelzer and Bailliecite journal
last =Schmelzer
first =Thomas
coauthors= Robert Baillie
year =2008
month =June–July
title = Summing a Curious, Slowly Convergent Series
journal = American Mathematical Monthly
volume =115
issue = 6
pages = 525–540
publisher = Mathematical Association of America
location = Washington, DC
issn = 00029890 ] found an efficientalgorithm for the more general problem of any omitted string of digits (eg, 42).Convergence
Kempner's proof of convergence is simple and is repeated in many textbooks, for example Hardy and Wright [] Rp|120and Apostol. [] Rp|212We group the terms of the sum by the number of digits in the denominator. The number of "n"-digit positive integers that have no 9 digits is exactly 8(9"n"-1), and each is greater than or equal 10"n"-1, so the contribution of this group to the sum is less than 8(9/10)"n"-1. Therefore the whole sum is bounded by :.
The same argument works for any omitted digit. The result is also true if strings of "k" digits are omitted, for example if we omit all denominators that have a decimal substring of 42. This can be proved in almost the same way. First we observe that we can work with numbers in base 10k and omit all denominators that have the given string as a "digit". The analogous argument to the base 10 case shows that this series converges. Now switching back to base 10, we see that this series contains all denominators that omit the given string, as well as denominators that include it if it is not on a "k-digit" boundary. For example, if we are omitting 42, the base-100 series would omit 4217 and 1742, but not 1427, so it is larger than the series that omits all 42s.
Approximation methods
The series converges extremely slowly. Baillie remarks that after summing 1027 terms the remainder is still larger than 1.
The upper bound of 80 is very crude, and Irwin showed [cite journal
last = Irwin
first = Frank
year = 1916
month = May
title = A Curious Convergent Series
journal = American Mathematical Monthly
volume = 23
issue = 5
pages = 149–152
publisher = Mathematical Association of America
location = Washington, DC
issn = 00029890 ] by a slightly finer analysis of the bounds that the value of the Kempner series is between 22.4 and 23.3.
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