A082838 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 9 in base 10} 1/k.

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, 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, 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

Offset

2,1

Comments

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

References

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Links

Table of n, a(n) for n=2..106.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.

Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.

Aubrey J. Kempner, A Curious Convergent Series, American Mathematical Monthly, volume 21, number 2, February 1914, pages 48-50.  Or JSTOR.

Eric Weisstein's World of Mathematics, Kempner Series.

Wikipedia, Kempner series

Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series

Formula

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

Example

22.920676619264150348163657094375931914944762436998481568541998356... - Robert G. Wilson v, Jun 01 2009

Mathematica

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

Crossrefs

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

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

Sequence in context: A125313 A199058 A274569 * A074961 A359454 A300450

Adjacent sequences: A082835 A082836 A082837 * A082839 A082840 A082841

Keyword

nonn,cons,base

Author

Robert G. Wilson v, Apr 14 2003

Extensions

More terms from Robert G. Wilson v, Apr 14 2009

More terms from Robert G. Wilson v, Jun 01 2009

Minor edits by M. F. Hasler, Jan 13 2020

Status

approved