A082839 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 0 in base 10} 1/k.

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

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.

Sums of this type are now called Kempner series, cf. LINKS. Convergence of the series is not more surprising than, and related to the fact that almost all numbers are pandigital (these have asymptotic density 1), i.e., "almost no number lacks any digit": Only a fraction of (9/10)^(L-1) of the L-digit numbers don't have a digit 0. Using L-1 = [log_10 k] ~ log_10 k, this density becomes 0.9^(L-1) ~ k^(log_10 0.9) ~ 1/k^0.046. If we multiply the generic term 1/k with this density, we have a converging series with value zeta(1 - log_10 0.9) ~ 22.4. More generally, almost all numbers contain any given substring of digits, e.g., 314159, and the sum over 1/k becomes convergent even if we omit just the terms having 314159 somewhere in their digits. - M. F. Hasler, Jan 13 2020

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), 372-374.

Robert Baillie, Summing The Curious Series Of Kempner And Irwin, arXiv:0806.4410 [math.CA], 2008-2015. [Robert G. Wilson v, Jun 01 2009]

Frank Irwin, A Curious Convergent Series, Amer. Math. Monthly, 23 (1916), 149-152.

A. D. Wadhwa, An interesting subseries of the harmonic series, Amer. Math. Monthly, 78 (1975), 931-933.

A. D. Wadhwa, Some convergent subseries of the harmonic series, Amer. Math. Monthly, 85 (1978), 661-663.

Eric Weisstein's World of Mathematics, Kempner Series.

Wikipedia, Kempner series

Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) 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.

Cf. A052382 (zeroless numbers).

Sequence in context: A071501 A004572 A352799 * 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