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%I #31 Oct 17 2014 17:55:31
%S 8,6,8,8,5,7,4,2,4,8,8,7,8,8,3,9,2,4,5,2,9,7,8,1,4,6,2,0,7,8,6,7,3,6,
%T 5,5,1,7,9,8,0,5,9,8,6,2,5,4,8,6,0,9,4,5,5,1,5,5,2,6,0,9,6,7,7,6,7,7,
%U 9,6,9,3,6,8,1,9,2,6,6,8,4,1,3,6,7,2,9,6,4,6,2,8,0,6,1,6,8,5,4,3,9,7,9,3,6,2
%N Product_{k>=3} (1 - Pi^2/(2*k^2))*sec(Pi/k).
%D A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", 1986, Eq. 6.2.4.3, p. 757.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>
%e 0.868857424887883924529781462078673655179805986254860945515526...
%p evalf(product((1 - Pi^2/(2*k^2))*sec(Pi/k), k=3..infinity), 120) # _Vaclav Kotesovec_, Sep 19 2014
%t part1 = Product[(1 - Pi^2/(2*k^2)), {k, 3, Infinity}]; Block[{$MaxExtraPrecision = 1000}, Do[Print[N[part1/Exp[Sum[-(2^(2*n) - 1)/n*Zeta[2*n]*(Zeta[2*n] - 1 - 1/2^(2*n)), {n, 1, m}]], 130]], {m, 300, 350}]] (* _Vaclav Kotesovec_, Sep 20 2014 *)
%o (PARI) prodinf(k=3, (1 - Pi^2/(2*k^2))/cos(Pi/k)) \\ _Michel Marcus_, Sep 20 2014
%o (PARI) exp(sumpos(k=3,log((1-Pi^2/(2*k^2))/cos(Pi/k)))) \\ Converges much faster: 0.2s for 150 decimals (on i3@2.4GHz). - _M. F. Hasler_, Sep 20 2014
%Y Cf. A051762, A085365.
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_, Sep 19 2014
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