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%I #7 Oct 10 2013 04:15:05
%S 1,4,0,1,2,8,4,1,1,9,2,2,8,4,2,9,8,8,4,2,5,9,6,2,5,6,6,9,8,7,9,6,5,5,
%T 0,4,0,9,1,1,8,7,4,2,4,8,7,5,4,8,2,4,1,1,2,4,1,8,1,8,5,5,5,7,3,7,5,0,
%U 0,5,2,6,8,1,6,8,3,3,2,4,0,1,5,1,1,3
%N Decimal expansion of the upper limit of the convergents of the continued fraction [1, -1/2, 1/4, -1/8, ... ].
%C Since sum{(-1)*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.
%e u = 1.401284... = [1, 2, 2, 30, 1, 2, 1, 254, 1, 6, 1, 2046, 1, ...];
%e v = -0.48360... = [0, -2, -14, -1, -2, -1, -126, -1, -6, -1, ...].
%e Every term of the continued fraction of u is 1 or has the form -2 + 2^m; every term for v is -1 or has the form 2 - 2^m.
%t $MaxExtraPrecision = Infinity; z = 800; t = Table[((-1)^n) 2^(-n), {n, 0, z}]; u = N[Convergents[t][[z - 1]], 120]; v = N[Convergents[t][[z]], 120]; RealDigits[u] (* A229987 *)
%t RealDigits[v] (* A229988 *)
%Y Cf. A229988.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Oct 06 2013
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