A130552 -id:A130552 - cp's OEIS frontend

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

Wolfdieter Lang, Jul 13 2007

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.

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

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.

W. Lang, Rationals and limit.

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

A130557 Numerators of partial sums of a series for 6(5 - 4Zeta(3)).

Original entry on oeis.org

1, 10, 409, 10297, 8031, 394019, 9462581, 766743461, 8435956183, 1020884056543, 13272613316059, 2243198436149971, 2243285892433171, 2243347792046947, 305101392961615867, 88175602457796281563, 186150555360181760633

Offset: 2

Wolfdieter Lang, Jul 13 2007

Denominators are given in A130558.

The rational sequence r(n) = 24*Sum_{j=2..n} 1/(j^3*(j^2-1)), n >= 2, tends, in the limit n->infinity, to 6*(5-4*Zeta(3)) which is approximately 1.15063433.

			Rationals r(n), n >= 2: 1, 10/9, 409/360, 10297/9000, 8031/7000, 394019/343000, ....

Z. A. Melzak, Companion to concrete mathematics,( Vol.I), Wiley, New York, 1973, pp. 83-84.

W. Lang, Rationals and limit.

Cf. A130551/A130552 with the limit (4/5)*Zeta(3).

a(n) = numerator(r(n)), n >= 2, with the rationals r(n) defined above.

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

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A130557 Numerators of partial sums of a series for 6(5 - 4Zeta(3)).

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cp's OEIS Frontend

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

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A130557 Numerators of partial sums of a series for 6*(5 - 4*Zeta(3)).

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A130557 Numerators of partial sums of a series for 6(5 - 4Zeta(3)).