A082833 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 4 in base 10} 1/k.

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, 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, 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

Offset: 2

Author

Robert G. Wilson v, Apr 14 2003

Keywords

• nonn
• cons
• base

Comments

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

Examples

			21.32746579959003668663940148693951284375095170327002181725118954... - _Robert G. Wilson v_, Jun 01 2009
		

References

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

Links

• 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. [From _Robert G. Wilson v_, Jun 01 2009]
• Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]
• Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series [From _Robert G. Wilson v_, Jun 01 2009]

Crossrefs

Cf. A002387, A024101, A052406 (numbers with no 4), A011534 (numbers with a 4).

Cf. A082830, A082831, A082832, A082834, A082835, A082836, A082837, A082838, A082839 (analog for digits 1, 2, ..., 9 and 0).

Programs

Mathematica

(* see the Mmca in Wolfram Library Archive *) (* Robert G. Wilson v, Jun 01 2009 *)

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

Formula

Equals Sum_{k in A052406\{0}} 1/k, where A052406 = numbers with no digit 3. - M. F. Hasler, Jan 15 2020

Extensions

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