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A140502
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Decimal expansion of the sum of the series 1/9 + 1/19 + 1/29 + 1/39+ 1/49 + ... where the denominators have exactly one 9.
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4
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2, 3, 0, 4, 4, 2, 8, 7, 0, 8, 0, 7, 4, 7, 8, 4, 8, 3, 1, 9, 6, 7, 5, 9, 4, 9, 3, 0, 9, 7, 3, 6, 1, 7, 4, 8, 2, 5, 3, 8, 9, 5, 9, 2, 0, 3, 0, 6, 4, 7, 7, 3, 6, 2, 1, 3, 5, 5, 7, 8, 7, 8, 3, 0, 0, 8, 2, 6, 2, 0, 4, 2, 5, 7, 9, 2, 8, 0, 2, 6, 1, 0, 0, 7, 1, 4, 5, 6, 7, 1, 4, 8, 2, 1, 1, 8, 8, 3, 0, 7, 8, 2, 5, 7, 9
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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COMMENTS
| 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 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 23.04428708074784831968. Note that this is larger than the sum of Kempner's "no 9" series. We also show how to construct nontrivial subseries of the harmonic series that have arbitrarily large, but computable, sums. For example, the sum of 1/n where n has at most 434 occurrences of the digit 0 is about 10016.32364577640186109739.
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LINKS
| Robert Baillie, Summing the curious series of Kempner and Irwin
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EXAMPLE
| 23.04428708074784831968...
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MATHEMATICA
| (* first install irwinSums.m, see reference, then *) First@ RealDigits@ iSum[9, 1, 111] (* From Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 03 2010 *)
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CROSSREFS
| Sequence in context: A025638 A025639 A195827 * A175434 A154860 A132774
Adjacent sequences: A140499 A140500 A140501 * A140503 A140504 A140505
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KEYWORD
| cons,nonn,base
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 30 2008
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EXTENSIONS
| Offset corrected R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 26 2009
More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 03 2010
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