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Decimal expansion of Sum_{k>=0} 1 / (k^8 + 1).
2

%I #13 May 20 2022 08:31:04

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

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

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

%N Decimal expansion of Sum_{k>=0} 1 / (k^8 + 1).

%F Equals 1/2 + ((sqrt(2 + sqrt(2))*sinh(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sin(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 + sqrt(2))*Pi) - cos(sqrt(2 - sqrt(2))*Pi)) + (sqrt(2 + sqrt(2))*sin(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sinh(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 - sqrt(2))*Pi) - cos(sqrt(2 + sqrt(2))*Pi))) * Pi/8.

%F Equal 3/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(8*k)-1). - _Amiram Eldar_, May 20 2022

%e 1.504062133314799511292905417451127075245414363820351975458635357818812...

%p evalf(1/2 + ((sqrt(2 + sqrt(2))*sinh(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sin(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 + sqrt(2))*Pi) - cos(sqrt(2 - sqrt(2))*Pi)) + (sqrt(2 + sqrt(2))*sin(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sinh(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 - sqrt(2))*Pi) - cos(sqrt(2 + sqrt(2))*Pi))) * Pi/8, 100);

%t RealDigits[Chop[N[Sum[1/(k^8 + 1), {k, 0, Infinity}], 105]]][[1]]

%o (PARI) sumpos(k=0, 1/(k^8 + 1))

%Y Cf. A060890, A113319, A354051, A354052.

%K nonn,cons

%O 1,2

%A _Vaclav Kotesovec_, May 16 2022