Rationals r(n):=A130035(n)/A130036(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)).

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(1/2,n), n=0..20 are:

[1, 17/16, 1097/1024, 17577/16384, 4500937/4194304, 72018961/67108864, 4609266865/4294967296, 73748453881/68719476736, 75518458183369/70368744177664, 1208295478677929/1125899906842624, 77330912768811177/72057594037927936, 1237294612076514873/1152921504606846976, 316747421148616537009/295147905179352825856, 5067958740068059597769/4722366482869645213696, 324349359389501776687841/302231454903657293676544, 5189589750326018446579481/4835703278458516698824704, 21256559617425695966000192201/19807040628566084398385987584, 340104953879151492830379493241/316912650057057350374175801344, 21766717048270842921188869198049/20282409603651670423947251286016, 348267472772353006858913963269409/324518553658426726783156020576256, 89156473029723557359976207290130729/83076749736557242056487941267521536]



The numerators A130035 are, for n=0..20:

[1, 17, 1097, 17577, 4500937, 72018961, 4609266865, 73748453881, 75518458183369, 1208295478677929, 77330912768811177, 1237294612076514873, 316747421148616537009, 5067958740068059597769, 324349359389501776687841, 5189589750326018446579481, 21256559617425695966000192201, 340104953879151492830379493241, 21766717048270842921188869198049, 348267472772353006858913963269409, 89156473029723557359976207290130729]


The denominators A130036 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]



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

[1.062500000, 1.073181998, 1.073182007, 1.073182007]
 

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

1.073182007 (maple10, 10 digits).


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