A082838 - OEIS

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

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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

From Amiram Eldar, Feb 01 2026: (Start)

Kempner (1914) proved that the series converges to a sum that is smaller than 90.

Irwin (1916) found that the sum is between 22.4 and 23.3.

Baillie (1979) calculated this constant to 20 decimal places, Fischer (1993) to 94 decimal places, and Schmelzer and Baillie (2008) to 100 decimal places. (End)

REFERENCES

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

LINKS

Amiram Eldar, Table of n, a(n) for n = 2..1003 (calculated using Baillie and Schmelzer's KempnerSums.nb)

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, Vol. 86, No. 5 (1979), 372-374.

Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2024.

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

Hans-Jürgen Fischer, Die Summe der Reziproken der natürlichen Zahlen ohne Ziffer 9, Elemente der Mathematik, Vol. 48, No. 3 (1993), pp. 100-106.

Frank Irwin, A Curious Convergent Series, The American Mathematical Monthly, Vol. 23, No. 5 (1916), pp. 149-152.

Aubrey J. Kempner, A Curious Convergent Series, American Mathematical Monthly, Volume 21, Number 2 (February 1914), pages 48-50. Or JSTOR.

Thomas Schmelzer and Robert Baillie, Summing a Curious, Slowly Convergent Series, The American Mathematical Monthly, Vol. 115, No. 6 (2008), pp. 525-540.

Eric Weisstein's World of Mathematics, Kempner Series.

Wikipedia, Kempner 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 code 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