Rationals r(n):=A120784(n)/A120785(n).

 r(n):=sum(C(k)/16^k,k=0..n) with C(k):=A000108(k)  (Catalan numbers).

 For n=0..30:
  
   [1, 17/16, 137/128, 4389/4096, 35119/32768, 561925/524288, 4495433/4194304, 287708141/268435456, 2301665843/2147483648, 
    36826655919/34359738368, 294613251551/274877906944, 9427624079025/8796093022208, 75420992684203/70368744177664, 
    1206735883132973/1125899906842624, 9653887065398089/9007199254740992, 1235697544380650237/1152921504606846976, 
    9885580355062880731/9223372036854775808, 158169285681070914091/147573952589676412928, 
    1265354285448686722403/1180591620717411303424, 40491337134358858748491/37778931862957161709568, 
    323930697074872511018033/302231454903657293676544, 5182891153197966292855283/4835703278458516698824704, 
    41463129225583741778162719/38685626227668133590597632, 2653640270437359645332220841/2475880078570760549798248448, 
    21229122163498877485133803559/19807040628566084398385987584, 
    339665954615982040977627457307/316912650057057350374175801344, 
    2717327636927856330116938792475/2535301200456458802993406410752, 
    86954484381691402581125429088201/81129638414606681695789005144064, 
    695635875053531220681971926674403/649037107316853453566312041152512, 
    11130174000856499531036831103871869/10384593717069655257060992658440192, 
    89041392006851996248533085487355721/83076749736557242056487941267521536]


  
 The values of some partial sums r(n) of the convergent series sum(C(k)/16^k,k=0..infty) are (maple10 10 digits):

   [1.062500000, 1.071796765, 1.071796770, 1.071796770], k=0,...,3. 


 This series is convergent (due to the quotient criterion). The limit is 4*(2-sqrt(3))=  1.071796769... 
 from the convergent Taylor expansion of sqrt(1+x) for the value x=-1/4  (the radius of convergence is R=1 due to the 
 quotient criterion). 
 The Lagrange remainder sequence for all |x|<1 tends to zero because 0<=|R(n,x)| < (1/2)*(C(n)/4^n) |x|^{n+1}, and
 (C(k)/4^k)*|x|^(k+1) is a 0-sequence for |x|<1  because the power series having these coefficients has radius of 
 convergence R=1. 


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