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Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of sqrt(2)/2 and 1.
3

%I #26 Jan 07 2024 08:26:10

%S 1,9,297,2401,308553,2472393,79169937,633543537,324415700169,

%T 2595473345377,83057280475785,664466019342321,85052107504546609,

%U 680418550231378497,21773418753366542529,174187444016951914257

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

%C The denominators are found in A130034.

%C The rationals r(n)=a(n)/A130034(n) (in lowest terms) converge for n->infinity to 1/agM(1,sqrt(2)/2). The value for sqrt(2)/2 is approx. 0.707.

%C 1/agM(1,sqrt(2)/2) approx. 1.180340599 multiplies 2*Pi*sqrt(l/g) to give the period T of a (mathematical) pendulum with maximal deflection of 90 degrees from the downward vertical. The length of the pendulum is l and g is the gravitational acceleration on the earth's surface, approx. 9.80665 m/s^2.

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

%D 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).

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

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 591, 17.3.11.

%H D. A. Cox, <a href="https://doi.org/10.5169/seals-53831">The arithmetic-geometric mean of Gauss</a>, L'Enseignement Mathématique, 30 (1984), 275-330.

%H Wolfdieter Lang, <a href="/A129934/a129934.txt">Rationals and limit.</a>

%F a(n) = numer( sum((((2*j)!/(j!^2))^2)*(1/2^(5*j)),j=0..n)), n>=0.

%F a(n) = numer(1+sum(((2*j-1)!!/(2*j)!!)^2*(1/2)^j,j=1..n)), n>=0, with the double factorials A001147 and A000165.

%e Rationals r(n) = [1, 9/8, 297/256, 2401/2048, 308553/262144, 2472393/2097152, ...]

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 01 2007