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%I #10 Oct 10 2013 09:18:45
%S 3,6,6,8,0,3,4,6,5,5,6,2,7,6,8,0,7,7,6,6,6,1,4,8,5,5,7,0,4,1,6,0,1,5,
%T 5,9,1,4,6,3,6,5,4,1,4,0,6,7,5,5,7,2,1,9,8,4,9,8,0,6,6,7,4,1,0,1,2,8,
%U 8,1,1,9,5,9,1,8,5,6,1,0,2,0,3,8,4,7
%N Decimal expansion of the upper limit of the convergents of the continued fraction [1, 1/3, 1/9, 1/27, ... ].
%C 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 u = 1.119... = [1, 8, 2, 1, 242, 8, 1, 6560, 26, 1, 177146, 80, 1,...];
%e v = 3.668... = [3, 1, 2, 80, 1, 8, 2186, 1, 26, 59048, 1, 80, ...].
%e In both cases, every term of the continued fraction has the form 3^m - 1.
%t $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 RealDigits[u] (* A229985 *)
%t RealDigits[v] (* A229986 *)
%Y Cf. A229985, A024023.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Oct 06 2013
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