A240264 Decimal expansion of Sum_{n >= 1} (-1)^(n+1)*H(n,2)/n^2, where H(n,2) is the n-th harmonic number of order 2.

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

Offset: 0

Author

Jean-François Alcover, Apr 03 2014

Keywords

• nonn
• cons

Examples

			0.631966197838...
		

References

• B. C. Berndt, Ramanujan's Notebooks Part I, Springer-Verlag,

Links

• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

Cf. A076788, A233090.

Programs

Mathematica

Zeta[3] - Pi^2/12*Log[2] // RealDigits[#, 10, 100]& // First

PARI

zeta(3)-log(2)*Pi^2/12 \\ Charles R Greathouse IV, Apr 03 2014

Formula

Equals zeta(3) - Pi^2/12*log(2).

Let a(p,q) = Sum_{n >= 1} (-1)^(n+1)*H(n,p)/n^q, then A076788 is a(1,1), A233090 is a(1,2) and this sequence is a(2,1).

Equals Sum_{n >= 1} (1/2)^n * H(n,1)/n^2, where H(n,1) = Sum_{k = 1..n} 1/k. See Berndt, p. 258. - Peter Bala, Oct 28 2021