A087491 Decimal expansion of the Khinchin harmonic mean K_{-1}.

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

Offset: 1

Author

Eric W. Weisstein, Sep 09 2003

Keywords

• nonn
• cons

Comments

Khinchin's constant is K_0 (A002210).

Examples

			1.7454056624073468634945963096836610672949366187...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.8, p. 61.

Links

• Eric Weisstein's World of Mathematics, Khinchin's Constant.
• Eric Weisstein's World of Mathematics, Khinchin Harmonic Mean.
• Wikipedia, Khinchin's constant.

Crossrefs

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

Programs

Mathematica

digits = 102; exactEnd = 1000; f[n_] = (1 - 1/(n + 1)^2)^(-1/n); s[n_] = Series[Log[f[n]], {n, Infinity, digits}] // Normal // N[#, digits] &; exactSum = Sum[Log[f[n]], {n, 1, exactEnd}] // N[#, digits] &; extraSum = Sum[s[n], {n, exactEnd + 1, Infinity}] // N[#, digits] &; A087491 = Log[2]/(exactSum + extraSum) // RealDigits // First  (* Jean-François Alcover, Feb 06 2013 *)
RealDigits[Log[2]/NSum[Log[(1 - 1/(n + 1)^2)^(-1/n)], {n, Infinity}, NSumTerms -> 10^4, WorkingPrecision -> 250, PrecisionGoal -> 110]][[1, ;; 100]] (* Eric W. Weisstein, Dec 10 2017 *)

Formula

Equals (Sum_{n>=1} -log2(1 - 1/(n+1)^2) * n^(-1))^(-1). - Jianing Song, Aug 08 2021