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%I #40 May 07 2021 12:26:33
%S 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,
%T 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,
%U 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
%N Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 0 in base 10} 1/k.
%C "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?"
%C "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.
%C 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
%D Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, p. 258.
%D Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, p. 34.
%D David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997.
%H Robert Baillie, <a href="http://www.jstor.org/stable/2321096">Sums of reciprocals of integers missing a given digit</a>, Amer. Math. Monthly, 86 (1979), 372-374.
%H Robert Baillie, <a href="https://arxiv.org/abs/0806.4410">Summing The Curious Series Of Kempner And Irwin</a>, arXiv:0806.4410 [math.CA], 2008-2015. [_Robert G. Wilson v_, Jun 01 2009]
%H Frank Irwin, <a href="http://www.jstor.org/stable/2974352">A Curious Convergent Series</a>, Amer. Math. Monthly, 23 (1916), 149-152.
%H A. D. Wadhwa, <a href="http://www.jstor.org/stable/2318503">An interesting subseries of the harmonic series</a>, Amer. Math. Monthly, 78 (1975), 931-933.
%H A. D. Wadhwa, <a href="http://www.jstor.org/stable/2320338">Some convergent subseries of the harmonic series</a>, Amer. Math. Monthly, 85 (1978), 661-663.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KempnerSeries.html">Kempner Series</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Kempner_series">Kempner series</a>
%H Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, <a href="http://library.wolfram.com/infocenter/MathSource/7166/"> Summing Kempner's Curious (Slowly-Convergent) Series</a> [_Robert G. Wilson v_, Jun 01 2009]
%e 23.10344790942054161603...
%t (* see the Mmca in Wolfram Library Archive. - _Robert G. Wilson v_, Jun 01 2009 *)
%Y Cf. A002387, A052386, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838.
%Y Cf. A052382 (zeroless numbers).
%K nonn,cons,base
%O 2,1
%A _Robert G. Wilson v_, Apr 14 2003
%E More terms from _Robert G. Wilson v_, Jun 01 2009
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