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A237841 Decimal expansion of Ramanujan's AGM Continued Fraction R(2) = R_1(2,2).

%I A237841 #28 Apr 05 2024 11:10:14
%S A237841 9,7,4,9,9,0,9,8,8,7,9,8,7,2,2,0,9,6,7,1,9,9,0,0,3,3,4,5,2,9,2,1,0,8,
%T A237841 4,4,0,0,5,9,2,0,2,1,9,9,9,4,7,1,0,6,0,5,7,4,5,2,6,8,2,5,1,2,8,5,8,7,
%U A237841 7,3,8,7,4,5,5,7,0,8,5,9,4,3,5,2,3,2,5,3,2,0,9,1,1,1,2,9,3,6,2,5
%N A237841 Decimal expansion of Ramanujan's AGM Continued Fraction R(2) = R_1(2,2).
%C A237841 Other closed form evaluations of R(p/q):
%C A237841 R(1/4) = Pi/2 - 4/3,
%C A237841 R(1/3) = 1 - log(2),
%C A237841 R(1/2) = 2 - Pi/2,
%C A237841 R(2/3) = 4 - Pi/sqrt(2),
%C A237841 R(1) = log(2),
%C A237841 R(3/2) = Pi + sqrt(3)*log(2 - sqrt(3)),
%C A237841 R(3) = Pi/sqrt(3) - log(2).
%H A237841 D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, <a href="http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf">Ten Problems in Experimental Mathematics</a>, page 13.
%H A237841 Jonathan M. Borwein, <a href="http://carmamaths.org/resources/jon/ag-talk10.pdf">Ramanujan's Arithmetic-Geometric Mean Continued Fractions and Dynamics</a>
%F A237841 Equivalent formulas:
%F A237841 sqrt(2)*(Pi/2 - log(1 + sqrt(2))),
%F A237841 (Pi - 2*arccoth(sqrt(2)))/sqrt(2),
%F A237841 Integral_{x >= 0} sech(Pi*x/4)/(1 + x^2) dx,
%F A237841 2*Integral_{x = 0..1} sqrt(x)/(1 + x^2) dx,
%F A237841 Integral_{x >= 0} exp(-x/2)*sech(x) dx,
%F A237841 4*Sum_{k >= 1} (-1)^(k+1)/(4*k - 1),
%F A237841 1/2*(-psi(3/8) + psi(7/8)), where psi is the digamma function,
%F A237841 4/3 * 2F1(3/4, 1, 7/4, -1), where 2F1 is the hypergeometric function,
%F A237841 (H(-1/8) - H(-5/8))/2, where H(n) is the n-th harmonic number.
%F A237841 General formula:
%F A237841 The Borwein's closed form formula for R(n) with n integer simplifies to:
%F A237841 R(n) = Pi/2*sec(Pi/(2n)) - 2*sum( cos((k*(n+1)*Pi)/(2*n))*log(2*sin((k*Pi)/(4*n))), {k, 1, 2n-1, 2} ).
%F A237841 Equals 4*A181049. - _Peter Bala_, Apr 02 2024
%e A237841 0.97499098879872209671990033452921084400592...
%t A237841 RealDigits[Sqrt[2]*(Pi/2 - Log[1 + Sqrt[2]]), 10, 100] // First
%o A237841 (PARI) (psi(7/8)-psi(3/8))/2 \\ _Charles R Greathouse IV_, Mar 03 2016
%Y A237841 Cf. A002162: R(1) = log(2); A180434: R(1/2) = 2-Pi/2.
%K A237841 nonn,cons
%O A237841 0,1
%A A237841 _Jean-François Alcover_, Feb 14 2014