A369633 Decimal expansion of integral of frac(1/x)^3 dx for x=0 to 1.

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, 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, 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

Offset: 0

Author

Benoit Cloitre, Jan 28 2024

Keywords

• nonn
• cons

Examples

			0.18707307250977978945095915767776663195781480296221...
		

References

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

Crossrefs

Cf. A074962, A345208.

Programs

Mathematica

RealDigits[3*Log[2*Pi]/2 - 6*Log[Glaisher] - EulerGamma - 1/2, 10, 120][[1]] (* Amiram Eldar, Jan 28 2024 *)

PARI

3*log(2*Pi)/2 + 6*zeta'(-1) - Euler - 1 \\ Amiram Eldar, Jan 28 2024

Formula

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.

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

From Vaclav Kotesovec, Jan 29 2024: (Start)

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

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