A130551 Numerators of partial sums for a series of (4/5)*Zeta(3).

1, 23, 1039, 58157, 1454021, 6854599, 30564710941, 244517610353, 37411196579209, 64619338818497, 86008340157931507, 8951094220597141, 334314418075511195849, 334314418069194908729, 48475590620225838341897

Offset: 1

Author

Wolfdieter Lang, Jul 13 2007

Keywords

• nonn
• frac
• easy

Comments

The rationals r(n):=2*sum(((-1)^(j-1))/((j^3)*binomial(2*j,j)),j=1..n), tend for n->infinity, to (4/5)*Zeta(3), which is approximately 0.9616455224. See the van der Poorten reference.

The denominators are given in A130552.

Examples

			Rationals r(n): [1, 23/24, 1039/1080, 58157/60480, 1454021/1512000, ...].
		

References

• A. van der Poorten, A proof that Euler missed..., Math. Intell. 1(1979)195-203; reprinted in Pi: A Source Book, pp. 439-447, eq. 2, with a proof in section 3 and further references in footnote 4.
• L. Berggren, T. Borwein and P. Borwein, Pi: A Source Book, Springer, New York, 1997, p. 687.

Links

• W. Lang, Rationals and limit.

Formula

a(n)=numerator(r(n)), n>=1, with the rationals r(n) defined above and taken in lowest terms.