A196521 Decimal expansion of Pi/4-log(2)/2.
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, 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, 7, 2, 9, 0, 5, 8, 6
Offset: 0
Examples
0.438824573117475654907044785090787437011542282663648828183396143330257...
References
- L. B. W. Jolley, Summation of series, Dover Publications Inc., New York, 1961, p. 14 (eq. 72).
Links
- Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Programs
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Maple
Digits:=100; evalf(Pi/4-log(2)/2); # Wesley Ivan Hurt, Dec 06 2013
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Mathematica
RealDigits[Pi/4 - Log[2]/2, 10, 100] (* Wesley Ivan Hurt, Dec 06 2013 *)
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PARI
Pi/4-log(2)/2 \\ Altug Alkan, Dec 14 2015
Formula
Equals 1 - 1/2 - 1/3 + 1/4 + 1/5 - ....
Equals Sum_{n>=0} 2/((4*n+2)*(4*n+3)). - Peter Luschny, Dec 06 2013
Equals Sum_{n>=1} (-1)^(n+1)/((2*n-1)*(2*n)). - Robert FERREOL, Dec 14 2015
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
Equals Integral_{x>=0} (exp(x) - 1)/(exp(2*x) + 1) dx. - Peter Bala, Nov 01 2019
From Bernard Schott, Sep 07 2020: (Start)
Equals Sum_{n>=1} (-1)^(n*(n-1)/2) / n [compare with A231902 formula].
Equals Sum_{n>=0} (8*n+5) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)
Equals Sum_{k>=1} A033264(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021
From Peter Bala, Mar 04 2025: (Start)
Equals (1/2) * A342316.
Equals Integral_{x = 0..1} x/(x^2 - 2*x + 2) = Integral_{x = 0..1} x*(1 + x)/(2 - x^2*(1 - x)) dx.
Equals (5/2)*Sum_{n >= 1} 1/(n*binomial(3*n, n)*2^n). The first 10 terms of the series gives the approximate value 0.43882457311(68...), correct to 11 decimal places. (End)