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%I #9 Jan 10 2014 16:23:48
%S 6,5,9,8,9,8,6,7,8,2,6,1,3,6,2,3,6,7,2,2,4,2,9,8,1,0,7,2,6,5,7,1,8,3,
%T 4,7,7,5,4,4,2,2,8,9,7,1,1,4,6,3,9,9,5,6,5,1,6,2,5,6,5,5,7,0,4,4,8,0,
%U 0,5,3,5,9,9,7,1,3,1,5,7,2,4,7,2,7,0,6,1,7,0,8,0,0,3,8,1,8,0,7,7,9,4,9,8,9
%N Decimal expansion of the lower limit of the convergents of the continued fraction [1/2, 1/4, 1/8, ... ].
%C Since sum{2^(-k), k=1,2,...} converges, the convergents of [1/2, 1/4, 1/8, ... ] 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 = 0.659898678... = [0, 1, 1, 1, 15, 1, 3, 127, 1, 7, 1023, 1,...];
%e v = 3.507873955... = [3, 1, 1, 31, 3, 1, 255, 7, 1, 2047, 15, ...].
%e In both cases, every term of the continued fraction has the form 2^m - 1.
%t $MaxExtraPrecision = Infinity; z = 500; t = Table[2^(-n), {n, 1, z}]; u = N[Convergents[t][[z - 1]], 120]; v = N[Convergents[t][[z]], 120];
%t RealDigits[u] (* A229983 *)
%t RealDigits[v] (* A229984 *)
%Y Cf. A229982.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Oct 06 2013
%E Offset corrected by and more terms from _Rick L. Shepherd_, Jan 09 2014
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