A130411 Numerator of partial sums of a series for 3*(Pi-3).

1, 2, 61, 44, 989, 6346, 51197, 36056, 4127401, 2057402, 189721879, 236723324, 1422382919, 20600649518, 10227626700773, 638723926928, 1278290544991, 23635180313246, 94585786464329, 969106771716436, 83372817133541471

Offset: 1

Author

Wolfdieter Lang, Jun 01 2007, Sep 09 2008, Oct 06 2008

Keywords

• nonn
• frac
• easy

Comments

Denominators are given in A130412.

The rationals (in lowest terms) r(n):=3*sum(((-1)^(j+1))/(j*(j+1)*(2*j+1)),j=1..n) have the limit 3*(Pi-3), approximately 0.424777962, for n->infinity.

These partial sums result from those for the more familiar series s(n):=sum(((-1)^(j+1))/(2*j*(2*j+1)*(2*j+2)),j=1..n) with limit (Pi-3)/4 which is approximately 0.0353981635. r(n)= 12*s(n). This series is attributed to K. G. Nilakantha, see, e.g., the R. Roy reference. eq.(13).

The sum r(n)/3 gives the n-th approximant to the continued fraction 1^2/(6+3^2/(6+5^2/6+...Proof with Euler's 1748 conversion of continued fractions into series. The denominators q(n)=A001879 of the n-th approximant of this continued fraction is used. The author (WL) reconsidered this entry after an e-mail from R. Rosenthal Jul 16 2008 pointing out the Pi-3 continued fraction.

Examples

			Rationals r(n), n>=1: [1/2, 2/5, 61/140, 44/105, 989/2310, 6346/15015, 51197/120120, ...].
Rationals s(n)=r(n)/12, n>=1: [1/24, 1/30, 61/1680, 11/315, 989/27720, 3173/90090, 51197/1441440, ...].
		

Links

• 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. Reprinted in: Pi: A Source Book, eds. L. Berggren, et al., Springer, New York, 1997, pp. 92-107.

Formula

a(n) = numerator(r(n)) with the rationals r(n) given above.