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A229981
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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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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, 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, 7, 4, 2, 4, 5, 8, 3, 1, 0, 9, 2, 1, 3, 6, 5, 4, 5, 9
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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u = 1.28507295... = [1, 3, 1, 1, 31, 3, 1, 255, 7, 1, 2047,...];
v = 2.51538415... = [2, 1, 1, 15, 1, 3, 127, 1, 7, 1023, 1, 15,...].
In both cases, every term of the continued fraction has the form 2^m - 1.
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MATHEMATICA
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$MaxExtraPrecision = Infinity; z = 600; t = Table[2^(-n), {n, 0, z}]; u = N[Convergents[t][[z - 1]], 120]; v = N[Convergents[t][[z]], 120];
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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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