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%I #20 Aug 01 2015 13:49:06
%S 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 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 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 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 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 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 Robert Baillie, <a href="http://arxiv.org/abs/0806.4410">Summing the curious series of Kempner and Irwin</a>
%e 23.04428708074784831968...
%t (* first install irwinSums.m, see reference, then *) First@ RealDigits@ iSum[9, 1, 111] (* _Robert G. Wilson v_, Aug 03 2010 *)
%Y Cf. A082838.
%K cons,nonn,base
%O 2,1
%A _Jonathan Vos Post_, Jun 30 2008
%E Offset corrected _R. J. Mathar_, Jan 26 2009
%E More terms from _Robert G. Wilson v_, Aug 03 2010