A233033 Decimal expansion of Sum_{n>=1} (-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

References

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 241, eq. (4.143).

Links

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

Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), pp. 15-35. See page 32.

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.52, pp. 309-310, eq. (4.85), section 5.52, p. 326, section 6.52, pp. 502-508.

Index entries for sequences related to harmonic numbers.

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).

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. A001008, A002805.

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,changed

Author

Jean-François Alcover, Dec 03 2013

Status

approved