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The '''Kempner series''' is a modification of the [[harmonic series (mathematics)|harmonic series]], formed by omitting all terms whose denominator expressed in base 10 contains a 9 digit. That is, it is the sum
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:<math>{\sum_{n=1}^\infty \ }' \frac{1}{n}</math>
where the prime indicates that ''n'' takes only values whose decimal expansion has no 9s. The series was first studied by [[Aubrey J. Kempner|A. J. Kempner]] in 1914.<ref name="Kempner 1914">
{{cite journal
| last = Kempner
| first = A. J.
|date=February 1914
| title = A Curious Convergent Series
| jstor = 2972074
| journal = [[American Mathematical Monthly]]
| volume = 21
| issue = 2
| pages = 48&ndash;50
| publisher = Mathematical Association of America
| location = Washington, DC
| issn = 0002-9890
| doi = 10.2307/2972074
}}</ref> The series is interesting because of the counter-intuitive result that, unlike the harmonic series, the Kempner series converges (Kempner showed this value was less than 80 and Baillie<ref name="Baillie 1979" /> showed that to 20 decimals, the actual sum is 22.92067 66192 64150 34816
{{OEIS|A082838}}).


Schmelzer and Baillie<ref name="Schmelzer and Baillie 2008">
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{{cite journal
| last1 =Schmelzer
| first1 =Thomas
| first2=Robert
| last2=Baillie
|date=June–July 2008
| title = Summing a Curious, Slowly Convergent Series
| journal = [[American Mathematical Monthly]]
| volume =115
| issue = 6
| pages = 525&ndash;540
| publisher = Mathematical Association of America
| location = Washington, DC
| issn = 0002-9890
| mr = 2416253
| jstor = 27642532
}}</ref> found an efficient [[algorithm]] for the more general problem of any omitted string of digits. For example, the sum of 1/''n'' where ''n'' has no "42" is about 228.44630 41592 30813 25415. Another, more striking, example: the sum of 1/''n'' where ''n'' has no occurrence of the digit string "314159" is about 2302582.33386 37826 07892 02376. (All values are rounded in the last decimal place).
 
==Convergence==
 
Kempner's proof of convergence<ref name="Kempner 1914" /> is simple and is repeated in many textbooks, for example Hardy and Wright<ref>{{cite book | last = Hardy | first = G. H. | coauthors = E. M. Wright | title = An Introduction to the Theory of Numbers | publisher = Clarendon Press | location = Oxford | year = 1979 | edition=5th | isbn = 0-19-853171-0 }}</ref>{{Rp|120}} and Apostol.<ref>{{cite book | last = Apostol | first = Tom | title = Mathematical Analysis | publisher = Addison&ndash;Wesley | location = Boston | year = 1974 | isbn = 0-201-00288-4 }}</ref>{{Rp|212}} We 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<sup>''n''&minus;1</sup>), and each is bigger than or equal to 10<sup>''n''&minus;1</sup>, so the contribution of this group to the sum is less than 8(9/10)<sup>''n''&minus;1</sup>. Therefore the whole sum is bounded by
 
:<math>8 \sum_{n=1}^\infty \left(\frac{9}{10}\right)^{n-1} = 80.</math>
 
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.<ref name="Schmelzer and Baillie 2008" /> First we observe that we can work with numbers in base 10<sup>''k''</sup> 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.
 
Farhi<ref>
{{cite journal
| last = Farhi
| first = Bakir
|date=December 2008
| title = A Curious Result Related to Kempner's Series
| journal = American Mathematical Monthly
| volume = 115
| issue = 10
| pages = 933&ndash;938
| publisher = Mathematical Association of America
| location = Washington, DC
| issn = 0002-9890
| mr = 2468554
}}</ref> considered generalized Kempner series, namely, the sums ''S''(''d'',&nbsp;''n'') of the reciprocals of the positive integers that have exactly ''n'' instances of the digit&nbsp;''d'' where 0&nbsp;≤&nbsp;''d''&nbsp;≤&nbsp;9 (so that the original Kempner series is ''S''(9,&nbsp;0)). He showed that for each ''d'' the sequence of values ''S''(''d'',&nbsp;''n'') for ''n''&nbsp;≥&nbsp;1 is decreasing and converges to 10&nbsp;ln&nbsp;10. Interestingly, the sequence is not in general decreasing starting with ''n''&nbsp;=&nbsp;0; for example, for the original Kempner series we have ''S''(9,&nbsp;0) ≈&nbsp;22.921&nbsp;<&nbsp;23.026 ≈&nbsp;10&nbsp;ln&nbsp;10&nbsp;<&nbsp;''S''(9,&nbsp;''n'') for&nbsp;''n''&nbsp;≥&nbsp;1.
 
==Approximation methods==
 
The series converges extremely slowly. Baillie<ref name="Baillie 1979" /> remarks that after summing 10<sup>27</sup> terms the remainder is still larger than 1.
 
The upper bound of 80 is very crude, and Irwin showed<ref>
{{cite journal
| last = Irwin
| first = Frank
|date=May 1916
| title = A Curious Convergent Series
| jstor = 2974352
| journal = American Mathematical Monthly
| volume = 23
| issue = 5
| pages = 149–152
| publisher = Mathematical Association of America
| location = Washington, DC
| issn = 0002-9890
| doi = 10.2307/2974352
}}</ref> by a slightly finer analysis of the bounds that the value of the Kempner series is between 22.4 and 23.3.
 
Baillie<ref name="Baillie 1979">
{{cite journal
| last =Baillie
| first =Robert
|date=May 1979
| title = Sums of Reciprocals of Integers Missing a Given Digit
| jstor =2321096
| journal = American Mathematical Monthly
| volume = 86
| issue = 5
| pages = 372–374
| publisher = Mathematical Association of America
| location = Washington, DC
| issn = 0002-9890
| doi =10.2307/2321096
}}</ref> developed a recursion that expresses the contribution from each ''k''+1-digit block in terms of the contributions of the ''k''-digit blocks for all choices of omitted digit. This permits a very accurate estimate with a small amount of computation.
 
==Name of this series==
 
Most authors do not name this series. The name "Kempner series" is used in MathWorld<ref>{{MathWorld|title=Kempner series|urlname=KempnerSeries}}</ref> and in Havil's book ''Gamma'' on the [[Euler–Mascheroni constant]].<ref>{{cite book | last = Havil | first = Julian | title = Gamma: Exploring Euler's Constant | publisher = Princeton University Press | location = Princeton | year = 2003 | isbn = 978-0-691-09983-5 }}</ref>{{Rp|31–33}}
 
==See also==
* [[small set (combinatorics)|small set]]
 
==Notes==
<references />
 
==External links==
* [http://eprints.maths.ox.ac.uk/1106/1/NA-06-17.pdf "Summing Curious, Slowly Convergent, Harmonic Subseries"]. Preprint of the paper by Thomas Schmelzer and Robert Baillie.
 
[[Category:Mathematical series]]
[[Category:Numerical analysis]]
[[Category:Articles containing proofs]]

Latest revision as of 16:13, 11 January 2015

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