A386738 Decimal expansion of Integral_{x=0..1} {1/x}^4 dx, where {} denotes fractional part.

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

Offset: 0

Author

Amiram Eldar, Aug 01 2025

Keywords

• nonn
• cons

Examples

			0.14553289487913287197745596494722440166566646379514...
		

Links

• Ovidiu Furdui, Problem 3366, Crux Mathematicorum, Vol. 34, No. 6 (2008), pp. 362 and 365; Solution to Problem 3366, by Chip Curtis, ibid., Vol. 35, No. 6 (2009), pp. 403-405.
• Huizeng Qin and Youmin Lu, Integrals of Fractional Parts and Some New Identities on Bernoulli Numbers, Int. J. Contemp. Math. Sciences, Vol. 6, No. 15 (2011), pp. 745-761.

Crossrefs

Cf. A001620 (gamma), A002117, A061444, A073002, A306016.

Cf. A153810 (m=1), A345208 (m=2), A345208 (m=3), this constant (m=4).

Programs

Mathematica

RealDigits[Log[2*Pi] - 2*EulerGamma - 1/3 + (Zeta[3]/2 + Zeta'[2])/Zeta[2], 10, 120][[1]]

PARI

log(2*Pi) - 2*Euler - 1/3 + (zeta(3)/2 + zeta'(2))/zeta(2)

Formula

Equals log(2*Pi) - 2*gamma - 1/3 + 3*zeta(3)/Pi^2 + 6*zeta'(2)/Pi^2.

In general, for m >= 2, Integral_{x=0..1} {1/x}^m dx = log(2*Pi) - m*gamma/2 - 1/(m-1) - Sum_{k=1..floor((m-2)/2)} (-1)^k * (m!/(m-2*k-1)!) * zeta(2*k+1) / (2^(2*k+1) * Pi^(2*k)) + 2 * Sum_{k=1..floor((m-1)/2)} (-1)^(k-1) * (m!/(m-2*k)!) * zeta'(2*k) / (2*Pi)^(2*k).