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%I #90 Sep 06 2024 05:04:51
%S 1,2,3,3,7,0,0,5,5,0,1,3,6,1,6,9,8,2,7,3,5,4,3,1,1,3,7,4,9,8,4,5,1,8,
%T 8,9,1,9,1,4,2,1,2,4,2,5,9,0,5,0,9,8,8,2,8,3,0,1,6,6,8,6,7,2,0,2,7,5,
%U 0,5,6,0,2,8,0,2,4,0,0,6,5,5,3,7,5,2,2,1,6,7,5,4,6,4,8,1,9,0,2,8,9,7,8,0,0
%N Decimal expansion of Pi^2/8.
%C According to Beckmann, Euler discovered the formula for this number as a sum of squares of reciprocals of odd numbers, along with similar formulas for Pi^2/6 and Pi^2/12. - _Alonso del Arte_, Apr 01 2013
%C Equals the asymptotic mean of the abundancy index of the odd numbers. - _Amiram Eldar_, May 12 2023
%D F. Aubonnet, D. Guinin and B.Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
%D Petr Beckmann, A History of Pi, 5th Ed. Boulder, Colorado: The Golem Press (1982): p. 153.
%D George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 122.
%D Calvin C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98.
%D L. B. W. Jolley, Summation of Series, Dover (1961).
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 54.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals 1 + 1/(2*3) + (1/3)*(1*2)/(3*5) + (1/4)*(1*2*3)/(3*5*7) + ... [Jolley eq 276]
%F Equals Sum_{k >= 1} 1/(2*k - 1)^2 [Clawson and Wells]. - _Alonso del Arte_, Aug 15 2012
%F Equals 2*(Integral_{t=0..1} sqrt(1 - t^2) dt)^2. - _Alonso del Arte_, Mar 29 2013
%F Equals Sum_{k >= 1} 2^k/(k^2*binomial(2*k, k)). - _Jean-François Alcover_, Apr 29 2013
%F Equals Integral_{x=0..1} log((1+x^2)/(1-x^2))/x dx. - _Bruno Berselli_, May 13 2013
%F Equals limit_{p->0} Integral_{x=0..Pi/2} x*tan(x)^p dx. [_Jean-François Alcover_, May 17 2013, after Boros & Moll p. 230]
%F Equals A002388/8 = A102753/4 = A091476/2. - _R. J. Mathar_, Oct 13 2015
%F Equals Integral_{x>=0} x*K_0(x)*K_1(x)dx where K are modified Bessel functions [Gradsteyn-Ryzhik 6.576.4]. - _R. J. Mathar_, Oct 22 2015
%F Equals (3/4)*zeta(2) = (3/4)*A013661. - _Wolfdieter Lang_, Sep 02 2019
%F From _Amiram Eldar_, Jul 17 2020: (Start)
%F Equals -Integral_{x=0..1} log(x)/(1 - x^2) dx = Integral_{x>=1} log(x)/(x^2-1) dx.
%F Equals -Integral_{x=0..oo} log(x)/(1 - x^4) dx.
%F Equals Integral_{x=0..oo} arctan(x)/(1 + x^2) dx. (End)
%F Equals Integral_{x=0..1} log(1+x+x^2+x^3)/x dx (Aubonnet). - _Bernard Schott_, Feb 04 2022
%e 1.23370055013616982735431137498451889191421242590509882830166867202...
%e 1 + 1/9 + 1/25 + 1/49 + 1/81 + 1/121 + 1/169 + 1/225 + ... - _Bruno Berselli_, Mar 06 2017
%t RealDigits[Pi^2/8, 10, 105][[1]] (* _Robert G. Wilson v_ *)
%o (PARI) Pi^2/8 \\ _Charles R Greathouse IV_, Dec 04 2016
%o (PARI) sumpos(n=1,(2*n-1)^-2) \\ _Charles R Greathouse IV_, Mar 02 2018
%Y Cf. A013661 (Pi^2/6), A002388, A102753, A091476.
%K cons,nonn,changed
%O 1,2
%A _Sam Alexander_, Oct 01 2005
%E More terms from _Robert G. Wilson v_, Oct 04 2005
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