A261069 Decimal expansion of J_5 = Integral_{0..Pi/2} x^5/sin(x) dx.
2, 6, 3, 4, 3, 1, 8, 2, 9, 0, 5, 1, 8, 7, 5, 5, 1, 6, 2, 2, 1, 0, 3, 1, 5, 9, 6, 1, 2, 8, 4, 0, 5, 5, 0, 5, 5, 9, 4, 0, 9, 3, 4, 3, 5, 8, 9, 3, 1, 5, 5, 5, 8, 4, 2, 1, 2, 3, 2, 1, 2, 3, 6, 9, 5, 8, 7, 1, 8, 0, 4, 6, 4, 0, 9, 5, 7, 1, 9, 1, 2, 7, 0, 2, 5, 2, 4, 0, 7, 0, 9, 7, 8, 2, 6, 6, 0, 5, 6, 2, 9, 8, 6
Offset: 1
Examples
2.634318290518755162210315961284055055940934358931555842123212369587...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- J. M. Borwein, I. J. Zucker and J. Boersma, The evaluation of character Euler double sums, The Ramanujan Journal, April 2008, Volume 15, Issue 3, pp 377-405, see p. 13.
Programs
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Mathematica
J5 = (5*Catalan*Pi^4)/8 - (29*I*Pi^6)/2016 - 30*I*Pi^2*PolyLog[4, -I] + 240*I*PolyLog[6, -I]; RealDigits[J5 // Re, 10, 103] // First RealDigits[NIntegrate[x^5/Sin[x],{x,0,Pi/2},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Aug 09 2023 *)
Formula
J_5 = (5*Catalan*Pi^4)/8 - (29*i*Pi^6)/2016 - 30*i*Pi^2*PolyLog(4, -i) + 240*i*PolyLog(6, -i).
Also equals (40*Pi^2*(32*Catalan*Pi^2 - PolyGamma(3, 1/4) + PolyGamma(3, 3/4)) + PolyGamma(5, 1/4) - PolyGamma(5, 3/4))/2048.