A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

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

Offset: 0

Author

Jean-François Alcover, Dec 04 2013, after the comment by Peter Bala about A233033

Keywords

• nonn
• cons

Examples

			0.7512855644747464283748363509446562442281164327128118011201697220886...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

Links

• 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 II (13)
• Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), page 32.
• Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C5.15)
• Michael I. Shamos, A catalog of the real numbers, (2007). See p. 561.

Crossrefs

Cf. A002117 (zeta(3)), A197070 (3*zeta(3)/4), A233091 (7*zeta(3)/8), A076788 (alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3), A233033 (alternating sum with denominator n^3).

Programs

Mathematica

RealDigits[ 5*Zeta[3]/8, 10, 100] // First

Formula

Equals 5*zeta(3)/8.

Equals -Integral_{x=0..1} (log(1+x)*log(1-x)/x)*dx. - Amiram Eldar, May 06 2023

Equals Sum_{m>=1} Sum_{n>=1} (-1)^(m-1)/(m*n*(m + n)) (see Finch). - Stefano Spezia, Nov 02 2024