Rationals r(n):=A130037(n)/A130038(n), n>=0.

r(k,n):= sum((((2*j)!/((2^(2*j))*j!^2))^2)*k^(2*j),j=0..n) for |k|<1 is
the partial sum for r(k)= limit(r(k,n),n->infty) = (2/Pi)*F(k,Pi/2) with 
the complete elliptic integral of the first kind F(k,Pi/2), called K(k^2) 
in the Abramowitz-Stegun handbook (A-St), p.591, 17.3.11. 

limit(r(k,n),n->infty)=1/agM(1,sqrt(1-k^2)).

In the pendulum problem k=sin(phi(0)/2), where phi(0) is the maximal deflection angle from the vertical. If phi(0)>Pi/2 then one thinks of the pendulum as a mass point on a stiff (massless) wire of length l.


See the D. A. Cox and A-St references given in A130035. 

The complete elliptic integral of the first kind is called K(m) in the A-St reference 
and it equals F(m^2,Pi/2) in the Cox reference.


In the pendulum problem k=sin(phi(0)/2), where phi(0) is the maximal deflection angle from the vertical.

k':=cos(phi(0)/2) appears in the eq. 1/agM(1,k') = (2/Pi)*K(k^2).

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The rationals r(n):=r(sqrt(3)/2,n), n=0..20 are:

[1, 19/16, 1297/1024, 21427/16384, 5584537/4194304, 90317059/67108864, 5819191945/4294967296, 93509568787/68719476736, 96025484363113/70368744177664, 1539315795453883/1125899906842624, 98642187446349841/72057594037927936, 1579652412024652483/1152921504606846976, 404633901283356405409/295147905179352825856, 6476837137305655553419/4722366482869645213696, 414637849146342799444441/302231454903657293676544, 6635554240391280655451731/4835703278458516698824704, 27183118338268247041624456201/19807040628566084398385987584, 434973847219097175313095937291/316912650057057350374175801344, 27840320422516159877175294862249/20282409603651670423947251286016, 445467814243440315039963429912859/324518553658426726783156020576256, 114043901365751054938812206193263929/83076749736557242056487941267521536]


The numerators A130037 are, for n=0..20:

 [1, 19, 1297, 21427, 5584537, 90317059, 5819191945, 93509568787, 96025484363113, 1539315795453883, 98642187446349841, 1579652412024652483, 404633901283356405409, 6476837137305655553419, 414637849146342799444441, 6635554240391280655451731, 27183118338268247041624456201, 434973847219097175313095937291, 27840320422516159877175294862249, 445467814243440315039963429912859, 114043901365751054938812206193263929]  

The denominators A130038 are, for n=0..20:

[1, 16, 1024, 16384, 4194304, 67108864, 4294967296, 68719476736, 70368744177664, 1125899906842624, 72057594037927936, 1152921504606846976, 295147905179352825856, 4722366482869645213696, 302231454903657293676544, 4835703278458516698824704, 19807040628566084398385987584, 316912650057057350374175801344, 20282409603651670423947251286016, 324518553658426726783156020576256, 83076749736557242056487941267521536]

These entries seem to be the same as the denominators A130036 of 1/agM(1,sqrt(3)/2).
 

The values for r(10^N), N=0,1,2,3, are (maple10, 10 digits):

[1.187500000, 1.368935346, 1.372880501, 1.372880501].

 
They should be compared with the value for 1/agM(1,sqrt(3)/2) which is 

1.372880501 (maple10, 10 digits).


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