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%I #45 Aug 15 2020 08:45:03
%S 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,
%T 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,
%U 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
%N Decimal expansion of Sum_{m>=1} (1/(2^m*m^2)).
%C Dilog function Li_2(1/2).
%D L. B. W. Jolley, Summation of Series, Dover (1961), eq. (116) on page 22 and eq. (360c) on page 68.
%D L. Lewin, Polylogarithms and Associated Functions, North Holland (1981), A2.1(4).
%H G. C. Greubel, <a href="/A076788/b076788.txt">Table of n, a(n) for n = 0..5000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>
%F Equals 1 - (1+1/2)/2 + (1+1/2+1/3)/3 - ... [Jolley].
%F Equals Pi^2/12 - 1/2*(log(2))^2 [Lewin]. - _Rick L. Shepherd_, Jul 21 2004
%F From _Amiram Eldar_, Aug 15 2020: (Start)
%F Equals Sum_{k>=1} (-1)^(k+1)*H(k)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
%F Equals Integral_{x=0..1} log(1+x)/(x*(1+x)) dx. (End)
%e 0.5822405264650125059026563201596801087441984748...
%t RealDigits[ PolyLog[2, 1/2] , 10, 105] // First (* _Jean-François Alcover_, Feb 20 2013 *)
%o (PARI) \p 200 dilog(1/2) Pi^2/12-1/2*(log(2))^2
%Y Cf. A001008, A002805.
%K nonn,cons
%O 0,1
%A _N. J. A. Sloane_, Jun 05 2003
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