A130037 Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of 1/2 and 1.

1, 19, 1297, 21427, 5584537, 90317059, 5819191945, 93509568787, 96025484363113, 1539315795453883, 98642187446349841, 1579652412024652483, 404633901283356405409, 6476837137305655553419, 414637849146342799444441

Offset

0,2

Comments

1/agM(1,1/2) approx. 1.372880501 multiplies 2*Pi*sqrt(l/g) to give the period T of a (mathematical) pendulum on a massless stiff wire of length l with maximal deflection of 120 degrees from the downward vertical. The gravitational acceleration on the earth's surface is g approx. 9.80665 m/s^2.

The denominators coincide with A130036.

The rationals r(n) = a(n)/A130036(n) (in lowest terms) converge for n->infinity to 1/agM(1,1/2).

1/agM(1,1/2) = (2/Pi)*K(3/4); complete elliptic integral of the first kind (see the Abramowitz-Stegun reference). K(3/4) = F(sqrt(3)/2,Pi/2) in the Cox reference.

References

D. A. Cox, The arithmetic-geometric mean of Gauss, in L. Berggren, J, Borwein, P. Borwein, Pi: A Source Book, Springer, 1997, pp. 481-536. eqs.(1.8) and (1.9).

L. D. Landau and E. M. Lifschitz: Lehrbuch der Theoretischen Physik, Band I, Mechanik, p. 30.

Links

Table of n, a(n) for n=0..14.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 591, 17.3.11.

D. A. Cox, The arithmetic-geometric mean of Gauss, L'Enseignement Mathématique, 30 (1984), 275-330.

Wolfdieter Lang, Rationals and limit

Formula

a(n) = numerator(Sum_{j=0..n} ((2*j)!/(j!^2))^2*((3/2^6)^j)), n >= 0.

a(n) = numerator(1 + Sum_{j=1..n} ((2*j-1)!!/(2*j)!!)^2*(3/4)^j), n >= 0, with the double factorials A001147 and A000165.

Crossrefs

Cf. A130035/A130036 rationals for deflection angle of 60 degrees.

Sequence in context: A253127 A078955 A107673 * A047910 A357230 A237429

Adjacent sequences: A130034 A130035 A130036 * A130038 A130039 A130040

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Jun 01 2007

Status

approved