Rationals r(n):=A120778(n)/A120777(n) (denominators coincide with A120777, conjecture)

 r(n):=sum(C(k)/4^k,k=0..n) with the Catalan numbers A000108. 

 For n=0..30:
  
 [1, 5/4, 11/8, 93/64, 193/128, 793/512, 1619/1024, 26333/16384, 53381/32768, 215955/131072, 
 436109/262144, 3518265/2097152, 7088533/4194304, 28539857/16777216, 57414019/33554432, 1846943453/1073741824, 
 3711565741/2147483648, 14911085359/8589934592, 29941580393/17179869184, 240416274739/137438953472, 
 482473579583/274877906944, 1936010885087/1099511627776, 3883457090629/2199023255552, 62306843256889/35184372088832, 
 124936162550609/70368744177664, 500960136802799/281474976710656, 1004216192739617/562949953421312, 
 8051112929645937/4503599627370496, 16135194353260669/9007199254740992,  64666057690124097/36028797018963968, 
 129570552036628963/72057594037927936]


  The values of some partial sums r(n) of the convergent series sum(C(k)/4^k,k=0..infty) are (maple10 10 digits):

  [1.250000000, 1.663623810, 1.887860948, 1.964339799] for r(10^k), k=0,...,3. 

 
  This series is convergent (due to J. L. Raabe's criterion (cf. A119951)). The limit is 2 from the convergent 
  Taylor expansion of sqrt(1+x) for the boundary value x=-1 (the radius of convergence is R=1 due to the quotient
  criterion). The Lagrange remainder sequence for this value  x=-1 tends to zero because 
  0<=|R(n,x=-1)| < (1/2)*C(n)/4^n, and C(k)/4^k is a 0-sequence because the series with these coefficients 
  converges (see above due to J. L. Raabe's criterion). 


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