A341908 Decimal expansion of Integral_{x=0..1} x/(exp(x)-1) dx.

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, 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, 6, 8, 4, 8, 8, 6, 2, 4, 9, 8, 0, 3, 7, 6, 8, 7, 7, 1, 1

Offset

0,1

References

Alvaro Meseguer, Fundamentals of Numerical Mathematics for Physicists and Engineers, Wiley, 2020, Chapter 4, exercise 12, p. 128.

John Michael Rassias, Geometry, Analysis, and Mechanics, World Scientific, 1994, p. 14.

Links

Table of n, a(n) for n=0..86.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 998.

Mathematics MI, Integral x/(e^x - 1) from 0 to 1, YouTube video, May 8 2021.

Voodooguru, Collaboration between Integration and Summation, Mathematical Meanderings, May 9 2021.

Eric Weisstein's World of Mathematics, Debye Functions.

Eric Weisstein's World of Mathematics, Dilogarithm.

Eric Weisstein's World of Mathematics, Polylogarithm.

Wikipedia, Debye function.

Wikipedia, Polylogarithm.

Wikipedia, Spence's function.

Formula

Equals D_1(1) = Sum_{k>=0} A120082(k)/A120083(k), where D_n(x) are the Debye functions.

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.

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.

Equals Sum_{k>=1} (1 - (k+1)*exp(-k))/k^2.

Example

0.77750463411224827641758654542571050719247729622900...

Maple

evalf(-dilog(exp(1))-1/2, 140);  # Alois P. Heinz, Jun 04 2021

Mathematica

RealDigits[PolyLog[2, 1-1/E], 10, 100][[1]]

Prog

(PARI) intnum(x=0, 1, x/(exp(x)-1)) \\ Michel Marcus, Jun 04 2021

Crossrefs

Cf. A027641, A027642, A120082, A120083.

Sequence in context: A051726 A139824 A019799 * A216159 A276615 A103983

Adjacent sequences: A341905 A341906 A341907 * A341909 A341910 A341911

Keyword

nonn,cons

Author

Amiram Eldar, Jun 04 2021

Status

approved