A369633 - OEIS

%I #20 Jan 29 2024 09:15:58

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

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

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

%N Decimal expansion of integral of frac(1/x)^3 dx for x=0 to 1.

%D Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Problems in Mathematical Analysis, Springer, 2013. See p. 100.

%F Integral_{x=0..1} frac(1/x)^3 dx = (3/2)*log(2*Pi) - 6*log(A) - gamma - 1/2 = 0.1870730725..., where A is the Glaisher-Kinkelin constant.

%F Equals 3*log(2) - 3/2 + 3 * Sum_{k>=1} ((-1)^k/(k+3))*(zeta(k+1)-1).

%F From _Vaclav Kotesovec_, Jan 29 2024: (Start)

%F Equals 6 * Sum_{k>=1} (zeta(k+1) - 1) / ((k+1)*(k+2)*(k+3)).

%F Equals -1/2 + 6 * Sum_{k>=2} zeta(k) / (k*(k+1)*(k+2)). (End)

%e 0.18707307250977978945095915767776663195781480296221...

%t RealDigits[3*Log[2*Pi]/2 - 6*Log[Glaisher] - EulerGamma - 1/2, 10, 120][[1]] (* _Amiram Eldar_, Jan 28 2024 *)

%o (PARI) 3*log(2*Pi)/2 + 6*zeta'(-1) - Euler - 1 \\ _Amiram Eldar_, Jan 28 2024

%Y Cf. A074962, A345208.

%K nonn,cons

%O 0,2

%A _Benoit Cloitre_, Jan 28 2024