cp's OEIS Frontend

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

%I A187832 #55 Feb 11 2025 12:21:28
%S A187832 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 A187832 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 A187832 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 A187832 Decimal expansion of integral from 1/2 to 1 of (1-x)/x dx.
%C A187832 Replacing 1/2 with any other number 0 < t < 1, the value of the integral is t - 1 - log(t).
%D A187832 J.-M. Monier, Cours, Analyse, Tome 4, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 4.3.14 pages 53 and 367.
%H A187832 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A187832 Equals log(2) - 1/2 = A002162 - 1/2.
%F A187832 Equals Sum_{k>=1} 1/((2k-1)*(2k)*(2k+1)). - _Bruno Berselli_, Mar 16 2014
%F A187832 From _Amiram Eldar_, Jul 28 2020: (Start)
%F A187832 Equals Sum_{k>=0} (-1)^k/(k+3).
%F A187832 Equals Sum_{k>=2} 1/(k * 2^k).
%F A187832 Equals Sum_{k>=2} 1/(4*k^2 - 2*k).
%F A187832 Equals Sum_{k>=2} (zeta(k) - 1)/2^k.
%F A187832 Equals Sum_{k>=1} zeta(2*k + 1)/2^(2*k + 1). (End)
%F A187832 From _Bernard Schott_, Nov 22 2021: (Start)
%F A187832 Equals Sum_{k>=1} (S(k) - log(2)) when S(k) = Sum_{m=1..k} (-1)^(m+1) / m.
%F A187832 Equals Integral_{x=0..1} x/(1+x)^2 dx. (End)
%F A187832 Equals Sum_{k,m>=1} (-1)^(k+m)/(k+m). - _Amiram Eldar_, Jun 09 2022
%F A187832 Equals Integral_{x = 0..1} Integral_{y = 0..1} x*y/(x + y)^2 dy dx. - _Peter Bala_, Dec 12 2022
%e A187832 0.193147180559945309417232121458176568075500134360255254120680009493393621969...
%p A187832 (evalf(log(2) - 1/2), 111); # _Bernard Schott_, Nov 25 2021
%t A187832 RealDigits[Log[2] - 1/2, 10, 111][[1]]
%o A187832 (PARI) log(2)-1/2 \\ _Charles R Greathouse IV_, Dec 27 2012
%Y A187832 Apart from the first digit the same as A002162.
%Y A187832 Cf. A239354: Sum_{k>=1} 1/((2k)*(2k+1)*(2k+2)).
%K A187832 nonn,cons,easy
%O A187832 0,2
%A A187832 _Robert G. Wilson v_, Dec 27 2012