A233033 Decimal expansion of sum_(n=1..infinity) (-1)^(n-1)*H(n)/n^3 where H(n) is the n-th harmonic number.

8, 5, 9, 2, 4, 7, 1, 5, 7, 9, 2, 8, 5, 9, 0, 6, 1, 5, 5, 3, 9, 9, 0, 9, 9, 3, 9, 4, 7, 5, 7, 5, 9, 9, 8, 0, 7, 1, 2, 8, 8, 4, 3, 5, 0, 8, 6, 0, 4, 1, 4, 9, 2, 6, 7, 6, 0, 5, 2, 0, 6, 8, 9, 7, 6, 6, 3, 8, 3, 4, 8, 1, 5, 3, 3, 4, 8, 9, 2, 3, 3, 0, 7, 1, 1, 3, 8, 8, 3, 8, 1, 5, 1, 8, 8, 4, 3, 0, 6, 0

Offset

0,1

Links

Table of n, a(n) for n=0..99.

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 32.

Formula

Equals 11*Pi^4/360 +1/12*Pi^2*log(2)^2 -log(2)^4/12 -2*Li4(1/2) -7/4*log(2)*zeta(3).

Also, equals 1/2*integral_{z=0..1} (log(z)^2*log(1+z)) / (z*(1+z)) dz.

Example

0.859247157928590615539909939475759980712884350860414926760520689766...

Mathematica

RealDigits[ 11*Pi^4/360 + 1/12*Pi^2*Log[2]^2 - Log[2]^4/12 - 2*PolyLog[4, 1/2] - 7/4*Log[2]*Zeta[3], 10, 100] // First

Prog

(PARI) 11*Pi^4/360 + Pi^2*log(2)^2/12 - log(2)^4/12 - 2*polylog(4, 1/2) - 7*log(2)*zeta(3)/4 \\ Charles R Greathouse IV, Aug 27 2014

Crossrefs

Cf. A076788 (same alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3).

Sequence in context: A100126 A330111 A335822 * A244810 A273985 A347195

Adjacent sequences: A233030 A233031 A233032 * A233034 A233035 A233036

Keyword

nonn,cons

Author

Jean-François Alcover, Dec 03 2013

Status

approved