A076788 Decimal expansion of Sum_{m>=1} (1/(2^m*m^2)).
5, 8, 2, 2, 4, 0, 5, 2, 6, 4, 6, 5, 0, 1, 2, 5, 0, 5, 9, 0, 2, 6, 5, 6, 3, 2, 0, 1, 5, 9, 6, 8, 0, 1, 0, 8, 7, 4, 4, 1, 9, 8, 4, 7, 4, 8, 0, 6, 1, 2, 6, 4, 2, 5, 4, 3, 4, 3, 4, 7, 0, 4, 7, 8, 7, 3, 1, 7, 1, 0, 4, 4, 0, 7, 1, 6, 8, 3, 2, 0, 0, 8, 1, 6, 8, 4, 0, 3, 1, 8, 5, 8, 7, 9, 1, 5, 8, 5, 7, 1, 8, 5, 6, 4, 4
Offset: 0
Examples
0.5822405264650125059026563201596801087441984748...
References
- Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 221.
- L. B. W. Jolley, Summation of Series, Dover (1961), eq. (116) on page 22 and eq. (360c) on page 68.
- L. Lewin, Polylogarithms and Associated Functions, North Holland (1981), A2.1(4).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..5000
- R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864 Table I (6).
- Eugène-Charles Catalan, Mémoire sur la transformation des séries et sur quelques intégrales définies, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
- Eric Weisstein's World of Mathematics, Dilogarithm
Programs
-
Mathematica
RealDigits[ PolyLog[2, 1/2] , 10, 105] // First (* Jean-François Alcover, Feb 20 2013 *)
-
PARI
\p 200 dilog(1/2)
-
PARI
Pi^2/12-1/2*(log(2))^2
-
PARI
lerchphi(.5,2,1)/2 \\ Charles R Greathouse IV, Jan 30 2025
Formula
Equals 1 - (1+1/2)/2 + (1+1/2+1/3)/3 - ... [Jolley].
Equals Pi^2/12 - 1/2*(log(2))^2 [Lewin]. - Rick L. Shepherd, Jul 21 2004
From Amiram Eldar, Aug 15 2020: (Start)
Equals Sum_{k>=1} (-1)^(k+1)*H(k)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
Equals Integral_{x=0..1} log(1+x)/(x*(1+x)) dx. (End)
From Peter Bala, Aug 18 2024: (Start)
Equals Integral_{x = 0..1} (log(2) - log(1 + x))/(1 - x) dx. See Catalan, Section 51, but note error in equation 94.
Note that Pi^2/12 + 1/2*(log(2))^2 = Integral_{x >= 1} log(1 + x)/(x*(1 + x)) dx. (End)
Comments