Rationals r(n):=A120780(n)/ A120781(n) = sum(C(k)/8^k,k=0..n),
 
 with C(k):=A000108(k) (Catalan numbers)
 
 For n=0..30:
  
 [1, 9/8, 37/32, 597/512, 2395/2048, 19181/16384, 76757/65536, 2456653/2097152, 9827327/8388608, 78621047/67108864, 
  314488387/268435456, 5031843585/4294967296, 20127426343/17179869184, 161019596469/137438953472, 644078720181/549755813888, 
  41221047786429/35184372088832, 164884208824551/140737488355328, 1319073735418803/1125899906842624, 
  5276295061084887/4503599627370496, 84420721860989787/72057594037927936, 337682889084989253/288230376151711744, 
  2701463118796480779/2305843009213693952, 10805852486621243571/9223372036854775808, 345787279743409601097/295147905179352825856, 
  1383149119296114441219/1180591620717411303424, 11065192955584402130115/9444732965739290427392, 
  44260771824633527654479/37778931862957161709568, 708172349211519830200665/604462909807314587353088, 
  2832689396879047814771455/2417851639229258349412352, 22661515175157662795253061/19342813113834066795298816, 
  90646060700869087837393013/77371252455336267181195264]


  The values of some partial sums r(n) of the convergent series sum(C(k)/8^k,k=0..infty) are (maple10 10 digits):

  [1.125000000, 1.171560537, 1.171572875, 1.171572875]  for r(10^k), k=0,...,3. 

 
  This series is convergent (due to the quotient criterion). The limit is 2*(2-sqrt(2))= 1.171572875..... 
  from the convergent Taylor expansion of sqrt(1+x) for the value x=-1/2  (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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