A102753 Decimal expansion of (Pi^2)/2.

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, 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, 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

Offset: 1

Author

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

Keywords

• cons
• nonn

Comments

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

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

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

Pi^2/2 is the squared side-length of a square with diagonal Pi. - Wesley Ivan Hurt, Jan 28 2022

Examples

			4.9348022005446793094172454999380755676568497036203953132066746881100\ 224112096026215008867018592761159120129568870115720388....
		

References

• J. Rivaud, Analyse, Séries, Equations différentielles, Mathématiques Supérieures et Spéciales, Premier Cycle Universitaire, Vuibert, 1981, Exercice 2, p. 135.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, 1986, p. 53.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• T. Amdeberhan, L. Medina and V. H. Moll, The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals, equation 2.39, arXiv:0705.2379 [math.CA], 2007.
• Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27
• Eric Weisstein's World of Mathematics, Hypersphere.
• Eric Weisstein's World of Mathematics, Trigamma Function.
• Wikipedia, Hypersphere.
• Wikipedia, Volume of an n-ball.
• Index entries for transcendental numbers

Crossrefs

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

Programs

Mathematica

RealDigits[Pi^2/2, 10, 111][[1]] (* Robert G. Wilson v, Dec 15 2005 *)

PARI

Pi^2/2 \\ Michel Marcus, Sep 04 2015

Formula

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

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

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

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

From Peter Bala, Nov 05 2019: (Start)

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

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

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

Pi^2/20 = 0.1 * Pi^2/2 = Sum_{k>=1} 1/A026424(k)^2. - Amiram Eldar, Aug 17 2020

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

Pi^2/2 = Integral_{x = -oo..oo} x/sinh(x) dx (see Rivaud reference). - Bernard Schott, Jan 28 2022