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Decimal expansion of Pi/4-log(2)/2.
6

%I #53 Oct 24 2024 02:21:04

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

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

%U 7,2,9,0,5,8,6

%N Decimal expansion of Pi/4-log(2)/2.

%D L. B. W. Jolley, Summation of series, Dover Publications Inc., New York, 1961, p. 14 (eq. 72).

%H Jean-Paul Allouche and Jeffrey Shallit, <a href="https://doi.org/10.1007/BFb0097122">Sums of digits and the Hurwitz zeta function</a>, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.

%F Equals 1 - 1/2 - 1/3 + 1/4 + 1/5 - ....

%F Equals Sum_{n>=0} 2/((4*n+2)*(4*n+3)). - _Peter Luschny_, Dec 06 2013

%F Equals Sum_{n>=1} (-1)^(n+1)/((2*n-1)*(2*n)). - _Robert FERREOL_, Dec 14 2015

%F Equals Integral_{x=0..1} (arctan(x)) dx = Integral_{x=0..Pi/4} (x / cos(x)^2) dx = Integral_{x=0..1/sqrt(2)} (arcsin(x)/(1-x^2)^(3/2)) dx. - _Robert FERREOL_, Dec 14 2015

%F Equals Integral_{x>=0} (exp(x) - 1)/(exp(2*x) + 1) dx. - _Peter Bala_, Nov 01 2019

%F From _Bernard Schott_, Sep 07 2020: (Start)

%F Equals Sum_{n>=1} (-1)^(n*(n-1)/2) / n [compare with A231902 formula].

%F Equals Sum_{n>=0} (8*n+5) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)

%F Equals Sum_{k>=1} A033264(k)/(k*(k+1)) (Allouche and Shallit, 1990). - _Amiram Eldar_, Jun 01 2021

%e 0.438824573117475654907044785090787437011542282663648828183396143330257...

%p Digits:=100; evalf(Pi/4-log(2)/2); # _Wesley Ivan Hurt_, Dec 06 2013

%t RealDigits[Pi/4 - Log[2]/2, 10, 100] (* _Wesley Ivan Hurt_, Dec 06 2013 *)

%o (PARI) Pi/4-log(2)/2 \\ _Altug Alkan_, Dec 14 2015

%Y Cf. A003881, A016655 (10*log(2)/2), A033264.

%Y Cf. A231902 (Pi/4+log(2)/2).

%K nonn,cons,easy,changed

%O 0,1

%A _R. J. Mathar_, Oct 03 2011