A187832 Decimal expansion of integral from 1/2 to 1 of (1-x)/x dx.

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

Offset: 0

Author

Robert G. Wilson v, Dec 27 2012

Keywords

• nonn
• cons
• easy

Comments

Replacing 1/2 with any other number 0 < t < 1, the value of the integral is t - 1 - log(t).

Examples

			0.193147180559945309417232121458176568075500134360255254120680009493393621969...
		

References

• J.-M. Monier, Cours, Analyse, Tome 4, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 4.3.14 pages 53 and 367.

Links

• Index entries for transcendental numbers.

Crossrefs

Apart from the first digit the same as A002162.

Cf. A239354: Sum_{k>=1} 1/((2k)*(2k+1)*(2k+2)).

Programs

Maple

(evalf(log(2) - 1/2), 111); # Bernard Schott, Nov 25 2021

Mathematica

RealDigits[Log[2] - 1/2, 10, 111][[1]]

PARI

log(2)-1/2 \\ Charles R Greathouse IV, Dec 27 2012

Formula

Equals log(2) - 1/2 = A002162 - 1/2.

Equals Sum_{k>=1} 1/((2k-1)*(2k)*(2k+1)). - Bruno Berselli, Mar 16 2014

From Amiram Eldar, Jul 28 2020: (Start)

Equals Sum_{k>=0} (-1)^k/(k+3).

Equals Sum_{k>=2} 1/(k * 2^k).

Equals Sum_{k>=2} 1/(4*k^2 - 2*k).

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

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

From Bernard Schott, Nov 22 2021: (Start)

Equals Sum_{k>=1} (S(k) - log(2)) when S(k) = Sum_{m=1..k} (-1)^(m+1) / m.

Equals Integral_{x=0..1} x/(1+x)^2 dx. (End)

Equals Sum_{k,m>=1} (-1)^(k+m)/(k+m). - Amiram Eldar, Jun 09 2022

Equals Integral_{x = 0..1} Integral_{y = 0..1} x*y/(x + y)^2 dy dx. - Peter Bala, Dec 12 2022