login
Decimal expansion of lim_{n -> infinity} Product_{k=1..2n} (1 - 2/(2*k+1))^(k*(-1)^k).
2

%I #14 Sep 08 2022 08:46:08

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

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

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

%N Decimal expansion of lim_{n -> infinity} Product_{k=1..2n} (1 - 2/(2*k+1))^(k*(-1)^k).

%H G. C. Greubel, <a href="/A241994/b241994.txt">Table of n, a(n) for n = 1..10000</a>

%H J.-P. Allouche, <a href="http://arxiv.org/abs/1305.6247">A note on products involving zeta(3) and Catalan's constant.</a> arXiv:1305.6247v3 [math.NT], 2014

%F exp(2*G/Pi - 1/2), where G is Catalan's constant (G = A006752 = 0.915965594...).

%e 1.08667416616077395213570672082096523329598330887030210204951841...

%t RealDigits[Exp[2*Catalan/Pi - 1/2] , 10, 105] // First

%o (PARI) default(realprecision, 100); exp(2*Catalan/Pi - 1/2) \\ _G. C. Greubel_, Aug 25 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:=RealField(); Exp(2*Catalan(R)/Pi(R) - 1/2); // _G. C. Greubel_, Aug 25 2018

%Y Cf. A006752, A241992, A241993.

%K nonn,cons,easy

%O 1,3

%A _Jean-François Alcover_, Aug 11 2014