Rationals r(n):=A120786(n)/A120787(n) = sum(C(k)/20^k,k=0..n).
 
 r(n):=sum(C(k)/20^k,k=0..n) with C(k):=A000108(k) (Catalan numbers).
 
For n=0..30:
  
 [1, 21/20, 211/200, 1689/1600, 84457/80000, 1689161/1600000, 16891643/16000000, 1351331869/1280000000, 2702663881/2560000000, 
  270266390531/256000000000, 2702663909509/2560000000000, 108106556409753/102400000000000, 1081065564149533/1024000000000000, 
  4324262256635277/4096000000000000, 43242622566419631/40960000000000000, 6918819610629079929/6553600000000000000, 
  69188196106294335057/65536000000000000000, 1383763922125899665619/1310720000000000000000, 
  22140222754014432861/20971520000000000000, 553505568850360998251319/524288000000000000000000, 
  5535055688503610310719211/5242880000000000000000000, 110701113770072207437697571/104857600000000000000000000, 
  1107011137700722076664039801/1048576000000000000000000000, 17712178203211553233485829089/16777216000000000000000000000, 
  4428044550802888308693933309081/4194304000000000000000000000000, 
  88560891016057766175094152781983/83886080000000000000000000000000, 
  885608910160577661753237446953849/838860800000000000000000000000000, 
  35424356406423106470146881265882961/33554432000000000000000000000000000, 
  70848712812846212940300356230559681/67108864000000000000000000000000000, 
  7084871281284621294030160903333049521/6710886400000000000000000000000000000, 
  70848712812846212940301847469986875979/67108864000000000000000000000000000000]

  
 The values of some partial sums r(n) of the convergent series sum(C(k)/20^k,k=0..infty) are (maple10 10 digits):

  [1.050000000, 1.055728090, 1.055728090, 1.055728090], k=0,...,3. 


 This series is convergent (due to the quotient criterion). The limit is 2*(5-2*sqrt(5))= (2*(7-4*phi)) = 1.055728090...,
 with the golden section phi:= (1+sqrt(5))/2. 
 Convergence follows from the Taylor expansion of sqrt(1+x) for the value x=-1/5  (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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