453:
digits are omitted, for example if we omit all denominators that have the decimal string 42. This can be proved in almost the same way. First we observe that we can work with numbers in base 10 and omit all denominators that have the given string as a "digit". The analogous argument to the base 10
239:−1 digits. Each of these numbers having no '9' is greater than or equal to 10, so the reciprocal of each of these numbers is less than or equal to 10. Therefore, the contribution of this group to the sum of reciprocals is less than 8 × (
230:
Kempner's proof of convergence is repeated in some textbooks, for example Hardy and Wright, and also appears as an exercise in
Apostol. We group the terms of the sum by the number of digits in the denominator. The number of
444:
334:
144:
Heuristically, this series converges because most large integers contain every digit. For example, a random 100-digit integer is very likely to contain at least one '9', causing it to be excluded from the above sum.
101:
454:
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 "
235:-digit positive integers that have no digit equal to '9' is 8 × 9 because there are 8 choices (1 through 8) for the first digit, and 9 independent choices (0 through 8) for each of the other
118:
because, unlike the harmonic series, it converges. Kempner showed the sum of this series is less than 90. Baillie showed that, rounded to 20 decimals, the actual sum is
540:
The upper bound of 80 is very crude. In 1916, Irwin showed that the value of the
Kempner series is between 22.4 and 23.3, since refined to the value above, 22.92067...
458:-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.
1173:
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33:
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509: ≥ 1 is decreasing and converges to 10 ln 10. The sequence is not in general decreasing starting with
24:
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The series converges extremely slowly. Baillie remarks that after summing 10 terms the remainder is still larger than 1.
1163:
1168:
664:
27:, formed by omitting all terms whose denominator expressed in base 10 contains the digit 9. That is, it is the sum
659:
563:-digit block. Therefore, with a small amount of computation, the original series (which is the value for
971:
837:
Schmelzer, Thomas; Baillie, Robert (June–July 2008). "Summing a
Curious, Slowly Convergent Series".
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1150:. Mathematica package by Thomas Schmelzer and Robert Baillie implementing their algorithm.
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517:(9, 0) ≈ 22.921 < 23.026 ≈ 10 ln 10 <
975:
110:
takes only values whose decimal expansion has no nines. The series was first studied by
152:
for the more general problem of any omitted string of digits. For example, the sum of
1157:
900:
791:
Baillie, Robert (May 1979). "Sums of
Reciprocals of Integers Missing a Given Digit".
1015:
884:
637:
one 9, is also convergent. Baillie showed that the sum of this last series is about
983:
852:
690:
1037:(10). Washington, DC: Mathematical Association of America: 866. December 1980.
952:
Farhi, Bakir (December 2008). "A Curious Result
Related to Kempner's Series".
559: + 1)-digit block in terms of all higher power contributions of the
1096:
1050:
991:
860:
812:
762:
695:
439:{\displaystyle 9\sum _{n=1}^{\infty }\left({\frac {9}{10}}\right)^{n-1}=90.}
329:{\displaystyle 8\sum _{n=1}^{\infty }\left({\frac {9}{10}}\right)^{n-1}=80.}
149:
1118:
Baillie, Robert (2008). "Summing the curious series of
Kempner and Irwin".
999:
868:
1104:
1058:
820:
770:
339:
The same argument works for any omitted non-zero digit. The number of
513: = 0; for example, for the original Kempner series we have
1088:
1042:
960:(10). Washington, DC: Mathematical Association of America: 933–938.
804:
754:
1083:(5). Washington, DC: Mathematical Association of America: 149–152.
847:(6). Washington, DC: Mathematical Association of America: 525–540.
799:(5). Washington, DC: Mathematical Association of America: 372–374.
96:{\displaystyle {\sideset {}{'}\sum _{n=1}^{\infty }}{\frac {1}{n}}}
1124:
966:
749:(2). Washington, DC: Mathematical Association of America: 48–50.
473:) of the reciprocals of the positive integers that have exactly
1144:. Preprint of the paper by Thomas Schmelzer and Robert Baillie.
739:
Kempner, A. J. (February 1914). "A Curious
Convergent Series".
16:
Harmonic series with all terms containing the digit '9' removed
461:
Farhi considered generalized
Kempner series, namely, the sums
343:-digit positive integers that have no '0' is 9, so the sum of
832:
830:
133:
583:
be a nonnegative integer. Irwin proved that the sum of 1/
1142:"Summing Curious, Slowly Convergent, Harmonic Subseries"
1075:
Irwin, Frank (May 1916). "A Curious
Convergent Series".
579:
In 1916, Irwin also generalized
Kempner's results. Let
209:
has no occurrence of the digit string "314159" is about
485: ≤ 9 (so that the original Kempner series is
1148:"Summing Kempner's Curious (Slowly-Convergent) Series"
222:. (All values are rounded in the last decimal place.)
374:
264:
255:). Therefore the whole sum of reciprocals is at most
36:
574:
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551:. He developed a recursion that expresses the
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906:(5th ed.). Oxford: Clarendon Press.
715:. Princeton: Princeton University Press.
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902:An Introduction to the Theory of Numbers
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489:(9, 0)). He showed that for each
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898:Hardy, G. H.; E. M. Wright (1979).
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547:-th powers simultaneously for all
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713:Gamma: Exploring Euler's Constant
555:-th power contribution from the (
1174:Base-dependent integer sequences
931:. Boston: Addison–Wesley.
571:) can be accurately estimated.
106:where the prime indicates that
984:10.1080/00029890.2008.11920611
853:10.1080/00029890.2008.11920559
187:. Another example: the sum of
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1077:American Mathematical Monthly
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954:American Mathematical Monthly
840:American Mathematical Monthly
793:American Mathematical Monthly
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477:instances of the digit
365:has no digit '0' is at most
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665:List of sums of reciprocals
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605:For example, the sum of 1/
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23:is a modification of the
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114:in 1914. The series is
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929:Mathematical Analysis
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226:Convergence
1158:Categories
1029:"ERRATA".
131:(sequence
1125:0806.4410
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1051:0002-9890
992:0002-9890
967:0807.3518
861:0002-9890
813:0002-9890
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696:MathWorld
660:Small set
521:(9,
423:−
395:∞
380:∑
313:−
285:∞
270:∑
176:228.44630
150:algorithm
78:∞
57:∑
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1016:34840740
1000:27642640
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654:See also
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1008:2468554
972:Bibcode
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