%I #53 Feb 08 2023 09:32:21

%S 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,

%T 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,

%U 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

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

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

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

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

%F Equals Sum_{k>=1} 1/((2k-1)*(2k)*(2k+1)). - _Bruno Berselli_, Mar 16 2014

%F From _Amiram Eldar_, Jul 28 2020: (Start)

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

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

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

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

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

%F From _Bernard Schott_, Nov 22 2021: (Start)

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

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

%F Equals Sum_{k,m>=1} (-1)^(k+m)/(k+m). - _Amiram Eldar_, Jun 09 2022

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

%e 0.193147180559945309417232121458176568075500134360255254120680009493393621969...

%p (evalf(log(2) - 1/2), 111); # _Bernard Schott_, Nov 25 2021

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

%o (PARI) log(2)-1/2 \\ _Charles R Greathouse IV_, Dec 27 2012

%Y Apart from the first digit the same as A002162.

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

%K nonn,cons,easy

%O 0,2

%A _Robert G. Wilson v_, Dec 27 2012