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

0,1

Comments

Dilog function Li_2(1/2).

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.

Konrad Knopp, Theory and application of infinite series, Blackie & Son Limited, London and Glasgow, 1954. See exercise 110 at page 269.

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.

Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 491-492.

Eric Weisstein's World of Mathematics, Dilogarithm.

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)

From Stefano Spezia, Feb 07 2026: (Start)

Equals Integral_{x=2..oo} log(x/(x - 1))/x = - Integral_{x=0..1} log((1 + x)/2)/(1 - x) [Shamos].

Equals - Integral_{x=0..r/2} log(1 - x/r)/x with r in R\{0}. (End)

Example

0.5822405264650125059026563201596801087441984748...

Mathematica

RealDigits[ PolyLog[2, 1/2] , 10, 105] // First (* Jean-François Alcover, Feb 20 2013 *)

Prog

(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

Crossrefs

Cf. A001008, A002805, A355234.

Sequence in context: A306068 A021949 A363540 * A195358 A021636 A011210

Adjacent sequences: A076785 A076786 A076787 * A076789 A076790 A076791

Keyword

nonn,cons

Author

N. J. A. Sloane, Jun 05 2003

Status

approved