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%I #25 Jan 16 2020 20:20:13
%S 2,1,3,2,7,4,6,5,7,9,9,5,9,0,0,3,6,6,8,6,6,3,9,4,0,1,4,8,6,9,3,9,5,1,
%T 2,8,4,3,7,5,0,9,5,1,7,0,3,2,7,0,0,2,1,8,1,7,2,5,1,1,8,9,5,4,1,9,7,7,
%U 8,8,4,2,7,2,4,5,1,3,3,5,3,7,5,3,8,1,2,0,1,3,0,2,8,4,0,6,9,3,5,4,7,7,8,9,7
%N Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 4 in base 10} 1/k.
%C Numbers with a digit 4 (A011534) have asymptotic density 1, i.e., here almost all terms are removed from the harmonic series, which makes convergence less surprising. See A082839 (the analog for digit 0) for more information about such so-called Kempner series. - _M. F. Hasler_, Jan 13 2020
%D Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.
%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="http://arxiv.org/abs/0806.4410">Summing the curious series of Kempner and Irwin</a>, arXiv:0806.4410 [math.CA], 2008-2015. [From _Robert G. Wilson v_, Jun 01 2009]
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Kempner_series">Kempner series</a>. [From _M. F. Hasler_, Jan 13 2020]
%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> [From _Robert G. Wilson v_, Jun 01 2009]
%F Equals Sum_{k in A052406\{0}} 1/k, where A052406 = numbers with no digit 3. - _M. F. Hasler_, Jan 15 2020
%e 21.32746579959003668663940148693951284375095170327002181725118954... - _Robert G. Wilson v_, Jun 01 2009
%t (* see the Mmca in Wolfram Library Archive *) (* _Robert G. Wilson v_, Jun 01 2009 *)
%o (PARI) sumpos(k=2,1/A052406(k)) \\ For illustration only, slow and not very precise: with \p19 takes 2 sec to get 5 digits right. - _M. F. Hasler_, Jan 13 2020
%Y Cf. A002387, A024101, A052406 (numbers with no 4), A011534 (numbers with a 4).
%Y Cf. A082830, A082831, A082832, A082834, A082835, A082836, A082837, A082838, A082839 (analog for digits 1, 2, ..., 9 and 0).
%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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