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

Author

Jean-François Alcover, Dec 03 2013

Keywords

• nonn
• cons

Examples

			0.859247157928590615539909939475759980712884350860414926760520689766...
		

Links

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

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).

Programs

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

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

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.