A130413 Numerators of partial sums for a series for Pi/3.

1, 19, 47, 1321, 989, 21779, 141481, 1132277, 801821, 91424611, 45706007, 4205393539, 5256312899, 31539920369, 457304942543, 226832956041173, 14176557010703, 28353956712541, 524535004412921, 2098185082863029

Offset: 0

Author

Wolfdieter Lang, Jun 01 2007

Keywords

• nonn
• frac
• easy

Comments

The denominators are given in A130414.

The rationals r(n) = 1 + (4/3)*Sum_{j=1..n} (-1)^(j+1)/((2*j+1)*((2*j+1)^2-1)), n >= 0, have the limit lim_{n->infinity} r(n) = Pi/3, approximately 1.047197551.

This series is obtained from the one for Pi/4 (attributed to Nilakantha) obtained by multiplication with 3/4. See the R. Roy link eq.(13).

Examples

			Rationals r(n): 1, 19/18, 47/45, 1321/1260, 989/945, 21779/20790, 141481/135135, ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1000
• W. Lang, Rationals and limit
• Ranjan Roy, The Discovery of the Series Formula for Pi by Leibniz, Gregory and Nilakantha, Math. Magazine 63 (1990), 291-306.

Crossrefs

Cf. A130411/A130412 (partial sums for a series of 3*(Pi-3)).

Programs

Maple

f:= n -> numer(1+ (4/3)*add(((-1)^(j+1))/((2*j+1)*((2*j+1)^2-1)),j=1..n)):
map(f, [$0..20]); # Robert Israel, Jul 27 2015

Formula

a(n) = numerator(r(n)), n >= 0, with r(n) defined above.

G.f. for r(n): 4*arctan(sqrt(x))/(3*sqrt(x)*(1-x)) - log(x+1)/(3*x). - Robert Israel, Jul 27 2015