A102753 - OEIS

%I #71 May 13 2024 09:18:55

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

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

%U 2,2,4,1,1,2,0,9,6,0,2,6,2,1,5,0,0,8,8,6,7,0,1,8,5,9,2,7,6,1,1,5,9,1,2,0,1

%N Decimal expansion of (Pi^2)/2.

%C Also equals the area under the peak-shaped even function f(x)=x/sinh(x).

%C Proof: For the upper half of the integral, write f(x) = 2x*exp(-x)/(1-exp(-2x)) = sum_{k=1..infinity} 2x*exp(-(2k-1)x) and integrate term by term from zero to infinity. - _Stanislav Sykora_, Nov 01 2013

%C Volume of the 4-dimensional unit sphere; the volume of the n-dimensional unit sphere is Pi^(n/2)/gamma(n/2+1) (see n-ball link and A164103). - _Rick L. Shepherd_, Jun 22 2017

%C Pi^2/2 is the squared side-length of a square with diagonal Pi. - _Wesley Ivan Hurt_, Jan 28 2022

%D J. Rivaud, Analyse, Séries, Equations différentielles, Mathématiques Supérieures et Spéciales, Premier Cycle Universitaire, Vuibert, 1981, Exercice 2, p. 135.

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, 1986, p. 53.

%H G. C. Greubel, <a href="/A102753/b102753.txt">Table of n, a(n) for n = 1..10000</a>

%H T. Amdeberhan, L. Medina and V. H. Moll, <a href="https://arxiv.org/abs/0705.2379">The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals</a>, equation 2.39, arXiv:0705.2379 [math.CA], 2007.

%H Renzo Sprugnoli, <a href="https://www.emis.de/journals/INTEGERS/papers/g27/g27.Abstract.html">Sums of reciprocals of the central binomial coefficients</a>, El. J. Combin. Numb. Th. 6 (2006) # A27

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hypersphere.html">Hypersphere</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrigammaFunction.html">Trigamma Function</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hypersphere">Hypersphere</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Volume_of_an_n-ball">Volume of an n-ball</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals psi_1(1/2), where psi_1(x) is the second logarithmic derivative of GAMMA(x).

%F Equals the volume of revolution of the sine or cosine curve for one half period, Integral_{0,Pi} Sin(x)^2 dx. - _Robert G. Wilson v_, Dec 15 2005

%F Equals Sum_{k >=1} 4^k/(k^2*binomial(2*k,k)) [Amdeberhan, Sprugnoli]. - _R. J. Mathar_, Sep 28 2007

%F Equals 4*Sum_{k >=1} 1/(2k-1)^2 [Wells].

%F From  _Peter Bala_, Nov 05 2019: (Start)

%F Pi^2/2 = Integral_{x = 0..inf} cosh(x)*x^2/sinh(x)^2 dx.

%F Pi^2/2 = 5*sum_{k >= 0} binomial(2*k,k)(-1/16)^k*1/(2*k+1)^2.

%F Pi^2/2 = 10*Integral_{x = 0..1/2}  1/x*log(x + sqrt(1 + x^2)) dx. (End)

%F Pi^2/20 = 0.1 * Pi^2/2 = Sum_{k>=1} 1/A026424(k)^2. - _Amiram Eldar_, Aug 17 2020

%F Conjecture: Pi^2/2 = Sum_{n = -oo..oo} ( cos(Pi*sqrt(n^2+1)) - cos(Pi*n) ) (using the Eisenstein summation convention). - _Peter Bala_, Oct 08 2021

%F Pi^2/2 = Integral_{x = -oo..oo} x/sinh(x) dx (see Rivaud reference). - _Bernard Schott_, Jan 28 2022

%e 4.9348022005446793094172454999380755676568497036203953132066746881100\ 224112096026215008867018592761159120129568870115720388....

%t RealDigits[Pi^2/2, 10, 111][[1]] (* _Robert G. Wilson v_, Dec 15 2005 *)

%o (PARI) Pi^2/2 \\ _Michel Marcus_, Sep 04 2015

%Y Cf. A002388, A000796, A019699, A026424, A164103, A164105, A164106, A164108, A248359, A276023.

%K cons,nonn

%O 1,1

%A Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 10 2005