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%I #14 Jun 04 2021 22:39:07
%S 7,7,7,5,0,4,6,3,4,1,1,2,2,4,8,2,7,6,4,1,7,5,8,6,5,4,5,4,2,5,7,1,0,5,
%T 0,7,1,9,2,4,7,7,2,9,6,2,2,9,0,0,0,8,6,9,1,7,9,4,9,4,5,4,1,0,6,9,9,6,
%U 6,8,4,8,8,6,2,4,9,8,0,3,7,6,8,7,7,1,1
%N Decimal expansion of Integral_{x=0..1} x/(exp(x)-1) dx.
%D Alvaro Meseguer, Fundamentals of Numerical Mathematics for Physicists and Engineers, Wiley, 2020, Chapter 4, exercise 12, p. 128.
%D John Michael Rassias, Geometry, Analysis, and Mechanics, World Scientific, 1994, p. 14.
%H M. Abramowitz and I. A. Stegun, eds., <a href="https://www.math.ubc.ca/~cbm/aands/page_998.htm">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 998.
%H Mathematics MI, <a href="https://www.youtube.com/watch?v=Ru5tQzg6jz8">Integral x/(e^x - 1) from 0 to 1</a>, YouTube video, May 8 2021.
%H Voodooguru, <a href="https://voodooguru23.blogspot.com/2021/05/collaboration-between-integration-and.html">Collaboration between Integration and Summation</a>, Mathematical Meanderings, May 9 2021.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DebyeFunctions.html">Debye Functions</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Debye_function">Debye function</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polylogarithm">Polylogarithm</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Spence%27s_function">Spence's function</a>.
%F Equals D_1(1) = Sum_{k>=0} A120082(k)/A120083(k), where D_n(x) are the Debye functions.
%F Equals Li_2(1-1/e) = -1/2 - Li_2(1-e) = Pi^2/6 - 1 + log(e-1) - Li_2(1/e), where Li_2(x) is the dilogarithm function.
%F Equals Sum_{k>=0} B(k)/(k+1)! = -1/2 + Sum_{k>=0} (-1)^k*B(k)/(k+1)! = -1/4 + Sum_{k>=0} B(2*k)/(2*k+1)!, where B(k) is the k-th Bernoulli number.
%F Equals Sum_{k>=1} (1 - (k+1)*exp(-k))/k^2.
%e 0.77750463411224827641758654542571050719247729622900...
%p evalf(-dilog(exp(1))-1/2, 140); # _Alois P. Heinz_, Jun 04 2021
%t RealDigits[PolyLog[2, 1-1/E], 10, 100][[1]]
%o (PARI) intnum(x=0, 1, x/(exp(x)-1)) \\ _Michel Marcus_, Jun 04 2021
%Y Cf. A027641, A027642, A120082, A120083.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Jun 04 2021
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