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Decimal expansion of Kempner series Sum_{k>=1, k has no digit 9 in base 10} 1/k.
17

%I #34 Jul 30 2020 04:14:19

%S 2,2,9,2,0,6,7,6,6,1,9,2,6,4,1,5,0,3,4,8,1,6,3,6,5,7,0,9,4,3,7,5,9,3,

%T 1,9,1,4,9,4,4,7,6,2,4,3,6,9,9,8,4,8,1,5,6,8,5,4,1,9,9,8,3,5,6,5,7,2,

%U 1,5,6,3,3,8,1,8,9,9,1,1,1,2,9,4,4,5,6,2,6,0,3,7,4,4,8,2,0,1,8,9,8,9,9,0,9

%N Decimal expansion of Kempner series Sum_{k>=1, k has no digit 9 in base 10} 1/k.

%C Numbers with a digit 9 (A011539) 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.

%H Aubrey J. Kempner, <a href="http://dx.doi.org/10.2307/2972074">A Curious Convergent Series</a>, American Mathematical Monthly, volume 21, number 2, February 1914, pages 48-50. Or <a href="https://www.jstor.org/stable/2972074">JSTOR</a>.

%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>

%F Equals Sum_{k in A007095\{0}} 1/k, where A007095 = numbers with no digit 9. - _M. F. Hasler_, Jan 15 2020

%e 22.920676619264150348163657094375931914944762436998481568541998356... - _Robert G. Wilson v_, Jun 01 2009

%t (* see the Mmca in Wolfram Library Archive link *)

%Y Cf. A002387, A007095 (numbers with no '9'), A011539 (numbers with a '9'), A024101.

%Y Cf. A082830 .. A082839 (analog for digits 1, ..., 8 and 0), A140502.

%K nonn,cons,base

%O 2,1

%A _Robert G. Wilson v_, Apr 14 2003

%E More terms from _Robert G. Wilson v_, Apr 14 2009

%E More terms from _Robert G. Wilson v_, Jun 01 2009

%E Minor edits by _M. F. Hasler_, Jan 13 2020