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A256358
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Decimal expansion of log(sqrt(Pi/2)).
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4
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2, 2, 5, 7, 9, 1, 3, 5, 2, 6, 4, 4, 7, 2, 7, 4, 3, 2, 3, 6, 3, 0, 9, 7, 6, 1, 4, 9, 4, 7, 4, 4, 1, 0, 7, 1, 7, 8, 5, 8, 9, 7, 3, 3, 9, 2, 7, 7, 5, 2, 8, 1, 5, 8, 6, 9, 6, 4, 7, 1, 5, 3, 0, 9, 8, 9, 3, 7, 2, 0, 7, 3, 9, 5, 7, 5, 6, 5, 6, 8, 2, 0, 8, 8, 8, 7, 9, 9, 7, 1, 6, 3, 9, 5, 3, 5, 5, 1, 0, 0, 8, 0, 0, 0, 4
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OFFSET
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0,1
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COMMENTS
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Equals the derivative of the Dirichlet eta function at x=0. - Stanislav Sykora, May 27 2015
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LINKS
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Costas Efthimiou, Problem 1838, Mathematics Magazine, Vol. 83, No. 1 (2010), p. 65; A weakly convergent series of logs, Solution to Problem 1838 by Tiberiu Trif, ibid., Vol. 84, No. 1 (2011), pp. 65-67.
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FORMULA
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Given the harmonic sum G(x) = Sum_{k>=1} (-1)^k*log(k)*exp(-k^2*x), lim_{x->0} G(x) = log(sqrt(Pi/2)).
Integral_{x=0..oo} G(x) dx = (Pi^2/12)*log(2) + zeta'(2)/2 = (Pi^2/12)*(EulerGamma + log(4*Pi) - 12*log(Glaisher)) = 0.1013165781635...
G'(0) = 7*zeta'(-2) = -7*zeta(3)/(4*Pi^2) = -0.2131391994...
Equals Integral_{-oo..+oo} -log(1/2 + i*z)/(exp(-Pi*z) + exp(Pi*z)) dz, where i is the imaginary unit. - Peter Luschny, Apr 08 2018
Equals Sum_{n>=0} Sum_{m>=1} (-1)^(m+n) * log(m+n)/(m+n) (Efthimiou, 2010). - Amiram Eldar, Apr 09 2022
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EXAMPLE
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0.22579135264472743236309761494744107178589733927752815869647153...
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MATHEMATICA
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RealDigits[Log[Sqrt[Pi/2]], 10, 105] // First
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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