1,2

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.

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.

W. Lang, Rationals and limit.

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

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,...].

Sequence in context: A121493 A078491 A101896 this_sequence A041449 A142729 A088078

Adjacent sequences: A130408 A130409 A130410 this_sequence A130412 A130413 A130414

nonn,frac,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jun 01 2007, Sep 09 2008, Oct 06 2008