Rationals r(n):=A129934(n)/A130034(n)

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 (A-St) handbook, p.591, 17.3.11. 

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

See the Cox and A-St references. 

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(2)/2,n), n=0..20 are:


 [1, 9/8, 297/256, 2401/2048, 308553/262144, 2472393/2097152, 79169937/67108864, 633543537/536870912, 324415700169/274877906944, 2595473345377/2199023255552, 83057280475785/70368744177664, 664466019342321/562949953421312, 85052107504546609/72057594037927936, 680418550231378497/576460752303423488, 21773418753366542529/18446744073709551616, 174187444016951914257/147573952589676412928, 356735975671126331036361/302231454903657293676544, 2853888145726385024708913/2417851639229258349412352, 91324425810624365372315841/77371252455336267181195264, 730595426005114815034627353/618970019642690137449562112, 93516215716258790557125463209/79228162514264337593543950336]


The numerators are A129934. For n=0..20:

[1, 9, 297, 2401, 308553, 2472393, 79169937, 633543537, 324415700169, 2595473345377, 83057280475785, 664466019342321, 85052107504546609, 680418550231378497, 21773418753366542529, 174187444016951914257, 356735975671126331036361, 2853888145726385024708913, 91324425810624365372315841, 730595426005114815034627353, 93516215716258790557125463209]

The denominators are A130034. For n=0..20:

 [1, 8, 256, 2048, 262144, 2097152, 67108864, 536870912, 274877906944, 2199023255552, 70368744177664, 562949953421312, 72057594037927936, 576460752303423488, 18446744073709551616, 147573952589676412928, 302231454903657293676544, 2417851639229258349412352, 77371252455336267181195264, 618970019642690137449562112, 79228162514264337593543950336]


Some values of r(n) (maple10, 10 digits) are, for r(10^N), N=0,1,2,3,

 [1.125000000, 1.180314946, 1.180340599, 1.180340599]

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

(maple10, 10 digits) 1.180340599.

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