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%I #7 Oct 09 2013 12:05:07
%S 1,2,8,5,0,7,2,9,5,6,6,6,2,4,3,1,9,8,3,2,0,3,9,2,2,7,0,6,5,1,7,9,7,1,
%T 4,3,8,8,1,4,4,0,1,5,4,6,4,7,7,9,0,6,6,6,1,2,5,9,6,2,0,5,2,7,7,9,6,0,
%U 7,4,2,4,5,8,3,1,0,9,2,1,3,6,5,4,5,9
%N Decimal expansion of the lower limit of the convergents of the continued fraction [1, 1/2, 1/4, 1/8, ... ].
%C Since sum{2^(-k), k=0,1,2,...} converges, the convergents of [1, 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; it appears that v = 1/(u-1).
%e u = 1.28507295... = [1, 3, 1, 1, 31, 3, 1, 255, 7, 1, 2047,...];
%e v = 2.51538415... = [2, 1, 1, 15, 1, 3, 127, 1, 7, 1023, 1, 15,...].
%e In both cases, every term of the continued fraction has the form 2^m - 1.
%t $MaxExtraPrecision = Infinity; z = 600; t = Table[2^(-n), {n, 0, z}]; u = N[Convergents[t][[z - 1]], 120]; v = N[Convergents[t][[z]], 120];
%t RealDigits[u] (* A229981 *)
%t RealDigits[v] (* A229982 *)
%Y Cf. A229982.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Oct 06 2013
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