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A140502 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.

%I A140502 #20 Aug 01 2015 13:49:06
%S A140502 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,
%T A140502 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,
%U A140502 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
%N A140502 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.
%C A140502 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 Irwin's series to high precision.
%C A140502 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.
%H A140502 Robert Baillie, <a href="http://arxiv.org/abs/0806.4410">Summing the curious series of Kempner and Irwin</a>
%e A140502 23.04428708074784831968...
%t A140502 (* first install irwinSums.m, see reference, then *) First@ RealDigits@ iSum[9, 1, 111] (* _Robert G. Wilson v_, Aug 03 2010 *)
%Y A140502 Cf. A082838.
%K A140502 cons,nonn,base
%O A140502 2,1
%A A140502 _Jonathan Vos Post_, Jun 30 2008
%E A140502 Offset corrected _R. J. Mathar_, Jan 26 2009
%E A140502 More terms from _Robert G. Wilson v_, Aug 03 2010