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A076788 Decimal expansion of Sum_{m>=1} (1/(2^m*m^2)).

%I A076788 #66 Feb 16 2025 08:32:47
%S A076788 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 A076788 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 A076788 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 A076788 Decimal expansion of Sum_{m>=1} (1/(2^m*m^2)).
%C A076788 Dilog function Li_2(1/2).
%D A076788 Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 221.
%D A076788 L. B. W. Jolley, Summation of Series, Dover (1961), eq. (116) on page 22 and eq. (360c) on page 68.
%D A076788 L. Lewin, Polylogarithms and Associated Functions, North Holland (1981), A2.1(4).
%H A076788 G. C. Greubel, <a href="/A076788/b076788.txt">Table of n, a(n) for n = 0..5000</a>
%H A076788 R. Barbieri, J. A. Mignaco, and E. Remiddi, <a href="https://dx.doi.org/10.1007/BF02728545">Electron form factors up to fourth order. I.</a>, Il Nuovo Cim. 11A (4) (1972) 824-864 Table I (6).
%H A076788 Eugène-Charles Catalan, <a href="https://www.persee.fr/doc/marb_0776-3123_1867_num_33_1_1705">Mémoire sur la transformation des séries et sur quelques intégrales définies</a>, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
%H A076788 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>
%F A076788 Equals 1 - (1+1/2)/2 + (1+1/2+1/3)/3 - ... [Jolley].
%F A076788 Equals Pi^2/12 - 1/2*(log(2))^2 [Lewin]. - _Rick L. Shepherd_, Jul 21 2004
%F A076788 From _Amiram Eldar_, Aug 15 2020: (Start)
%F A076788 Equals Sum_{k>=1} (-1)^(k+1)*H(k)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
%F A076788 Equals Integral_{x=0..1} log(1+x)/(x*(1+x)) dx. (End)
%F A076788 From _Peter Bala_, Aug 18 2024: (Start)
%F A076788 Equals Integral_{x = 0..1} (log(2) - log(1 + x))/(1 - x) dx. See Catalan, Section 51, but note error in equation 94.
%F A076788 Note that Pi^2/12 + 1/2*(log(2))^2 = Integral_{x >= 1} log(1 + x)/(x*(1 + x)) dx. (End)
%e A076788 0.5822405264650125059026563201596801087441984748...
%t A076788 RealDigits[ PolyLog[2, 1/2] , 10, 105] // First (* _Jean-François Alcover_, Feb 20 2013 *)
%o A076788 (PARI) \p 200 dilog(1/2)
%o A076788 (PARI) Pi^2/12-1/2*(log(2))^2
%o A076788 (PARI) lerchphi(.5,2,1)/2 \\ _Charles R Greathouse IV_, Jan 30 2025
%Y A076788 Cf. A001008, A002805, A355234.
%K A076788 nonn,cons
%O A076788 0,1
%A A076788 _N. J. A. Sloane_, Jun 05 2003