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

%I A087491 #27 Feb 16 2025 08:32:51
%S A087491 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,
%T A087491 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,
%U A087491 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
%N A087491 Decimal expansion of the Khinchin harmonic mean K_{-1}.
%C A087491 Khinchin's constant is K_0 (A002210).
%D A087491 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.8, p. 61.
%H A087491 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KhinchinsConstant.html">Khinchin's Constant</a>.
%H A087491 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KhinchinHarmonicMean.html">Khinchin Harmonic Mean</a>.
%H A087491 Wikipedia, <a href="http://en.wikipedia.org/wiki/Khinchin%27s_constant">Khinchin's constant</a>.
%F A087491 Equals (Sum_{n>=1} -log2(1 - 1/(n+1)^2) * n^(-1))^(-1). - _Jianing Song_, Aug 08 2021
%e A087491 1.7454056624073468634945963096836610672949366187...
%t A087491 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 *)
%t A087491 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 *)
%Y A087491 Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.
%K A087491 nonn,cons
%O A087491 1,2
%A A087491 _Eric W. Weisstein_, Sep 09 2003