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A082839 Decimal expansion of Sum_{ k >= 1, k has no zero digit in base 10 } 1/k. 13


%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 Sum_{ k >= 1, k has no zero digit 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.

%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 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.

%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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Last modified July 22 10:03 EDT 2019. Contains 325219 sequences. (Running on oeis4.)