This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A229985 #8 Oct 10 2013 09:18:59
%S A229985 1,1,1,9,9,9,3,4,0,9,9,7,2,9,5,8,7,4,0,9,1,4,2,8,3,2,4,8,2,6,0,9,5,3,
%T A229985 2,2,9,9,6,3,8,0,1,7,0,2,8,1,5,5,2,5,0,7,0,5,8,8,5,1,0,7,5,4,8,6,6,5,
%U A229985 4,1,5,4,6,4,6,4,2,7,4,9,8,8,2,5,8,4
%N A229985 Decimal expansion of the lower limit of the convergents of the continued fraction [1, 1/3, 1/9, 1/27, ... ].
%C A229985 Since sum{3^(-k), k = 0,1,2,...} converges, the convergents of [1, 1/3, 1/9, 1/27, ... ] diverge, by the Seidel Convergence Theorem. However, the odd-numbered convergents converge, as do the even-numbered convergents. In the Example section, these limits are denoted by u and v.
%e A229985 u = 1.119... = [1, 8, 2, 1, 242, 8, 1, 6560, 26, 1, 177146, 80, 1,...];
%e A229985 v = 3.668... = [3, 1, 2, 80, 1, 8, 2186, 1, 26, 59048, 1, 80, ...].
%e A229985 In both cases, every term of the continued fraction has the form 3^m - 1.
%t A229985 $MaxExtraPrecision = Infinity; z = 500; t = Table[3^(-n), {n, 0, z}]; u = N[Convergents[t][[z - 1]], 120]; v = N[Convergents[t][[z]], 120];
%t A229985 RealDigits[u] (* A229985 *)
%t A229985 RealDigits[v] (* A229986 *)
%Y A229985 Cf. A229986, A024023.
%K A229985 nonn,cons
%O A229985 1,4
%A A229985 _Clark Kimberling_, Oct 06 2013