A140502 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.

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, 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, 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

Offset

2,1

Comments

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.

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.

Links

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

Robert Baillie, Summing the curious series of Kempner and Irwin

Example

23.04428708074784831968...

Mathematica

(* first install irwinSums.m, see reference, then *) First@ RealDigits@ iSum[9, 1, 111] (* Robert G. Wilson v, Aug 03 2010 *)

Crossrefs

Cf. A082838.

Sequence in context: A286239 A341585 A343866 * A175434 A154860 A284282

Adjacent sequences: A140499 A140500 A140501 * A140503 A140504 A140505

Keyword

cons,nonn,base

Author

Jonathan Vos Post, Jun 30 2008

Extensions

Offset corrected R. J. Mathar, Jan 26 2009

More terms from Robert G. Wilson v, Aug 03 2010

Status

approved