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A229988
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Decimal expansion of the lower limit of the convergents of the continued fraction [1, -1/2, 1/4, -1/8, ... ].
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2
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4, 8, 3, 6, 0, 6, 0, 3, 0, 9, 2, 5, 2, 8, 9, 0, 8, 8, 9, 3, 9, 9, 2, 6, 3, 5, 0, 8, 5, 5, 9, 3, 9, 4, 8, 0, 7, 9, 0, 4, 2, 3, 5, 9, 0, 1, 6, 3, 2, 0, 8, 0, 4, 0, 6, 0, 9, 1, 1, 7, 8, 4, 4, 8, 6, 2, 1, 3, 7, 7, 2, 6, 4, 6, 0, 9, 6, 8, 4, 5, 2, 8, 2, 4, 1, 0
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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u = 1.401284... = [1, 2, 2, 30, 1, 2, 1, 254, 1, 6, 1, 2046, 1, ...];
v = -0.48360... = [0, -2, -14, -1, -2, -1, -126, -1, -6, -1, ...].
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.
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MATHEMATICA
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$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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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