OFFSET
0,2
REFERENCES
Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Problems in Mathematical Analysis, Springer, 2013. See p. 100.
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)
EXAMPLE
0.18707307250977978945095915767776663195781480296221...
MATHEMATICA
RealDigits[3*Log[2*Pi]/2 - 6*Log[Glaisher] - EulerGamma - 1/2, 10, 120][[1]] (* Amiram Eldar, Jan 28 2024 *)
PROG
(PARI) 3*log(2*Pi)/2 + 6*zeta'(-1) - Euler - 1 \\ Amiram Eldar, Jan 28 2024
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Benoit Cloitre, Jan 28 2024
STATUS
approved