

A082839


Decimal expansion of Sum_{ k >= 1, k has no zero digit in base 10 } 1/k.


12



2, 3, 1, 0, 3, 4, 4, 7, 9, 0, 9, 4, 2, 0, 5, 4, 1, 6, 1, 6, 0, 3, 4, 0, 5, 4, 0, 4, 3, 3, 2, 5, 5, 9, 8, 1, 3, 8, 3, 0, 2, 8, 0, 0, 0, 0, 5, 2, 8, 2, 1, 4, 1, 8, 8, 6, 7, 2, 3, 0, 9, 4, 7, 7, 2, 7, 3, 8, 7, 5, 0, 7, 9, 6, 0, 6, 1, 4, 1, 9, 4, 2, 6, 3, 5, 9, 2, 0, 1, 9, 1, 0, 5, 2, 6, 1, 3, 9, 3, 3, 8, 6, 5, 2, 1
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OFFSET

2,1


COMMENTS

"The most novel culling of the terms of the harmonic series has to be due to A. J. Kempner, who in 1914 considered what would happen if all terms are removed from it which have a particular digit appearing in their denominators. For example, if we choose the digits 7, we would exclude the terms with denominators such as 7, 27, 173,33779, etc. There are 10 such series, each resulting from the removal of one of the digits 0, 1, 2, ..., 9 and the first question which naturally arises is just what percentage of the terms of the series are we removing by the process?"
"The sum of the reciprocals, 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... [A002387] is unbounded. By taking sufficiently many terms, it can be made as large as one pleases. However, if the reciprocals of all numbers that when written in base 10 contain at least one 0 are omitted, then the sum has the limit, 23.10345... [Boas and Wrench, AMM v78]."  Wells.


REFERENCES

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, p. 258.
Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, p. 34.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997.


LINKS

Table of n, a(n) for n=2..106.
Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372374.
Robert Baillie, Summing The Curious Series Of Kempner And Irwin, arXiv:0806.4410 [math.CA], 20082015. [Robert G. Wilson v, Jun 01 2009]
Frank Irwin, A Curious Convergent Series, Amer. Math. Monthly, 23 (1916), 149152.
A. D. Wadhwa, An interesting subseries of the harmonic series, Amer. Math. Monthly, 78 (1975), 931933.
A. D. Wadhwa, Some convergent subseries of the harmonic series, Amer. Math. Monthly, 85 (1978), 661663.
Wolfram Library Archive, KempnerSums.nb (8.6 KB)  Mathematica Notebook, Summing Kempner's Curious (SlowlyConvergent) Series [Robert G. Wilson v, Jun 01 2009]


EXAMPLE

23.10344790942054161603...


MATHEMATICA

(* see the Mmca in Wolfram Library Archive *) (* Robert G. Wilson v, Jun 01 2009 *)


CROSSREFS

Cf. A002387, A052386, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838.
Sequence in context: A154720 A071501 A004572 * A130717 A137396 A244213
Adjacent sequences: A082836 A082837 A082838 * A082840 A082841 A082842


KEYWORD

nonn,cons,base


AUTHOR

Robert G. Wilson v, Apr 14 2003


EXTENSIONS

More terms from Robert G. Wilson v, Jun 01 2009


STATUS

approved



