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A229985
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Decimal expansion of the lower limit of the convergents of the continued fraction [1, 1/3, 1/9, 1/27, ... ].
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2
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1, 1, 1, 9, 9, 9, 3, 4, 0, 9, 9, 7, 2, 9, 5, 8, 7, 4, 0, 9, 1, 4, 2, 8, 3, 2, 4, 8, 2, 6, 0, 9, 5, 3, 2, 2, 9, 9, 6, 3, 8, 0, 1, 7, 0, 2, 8, 1, 5, 5, 2, 5, 0, 7, 0, 5, 8, 8, 5, 1, 0, 7, 5, 4, 8, 6, 6, 5, 4, 1, 5, 4, 6, 4, 6, 4, 2, 7, 4, 9, 8, 8, 2, 5, 8, 4
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OFFSET
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1,4
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COMMENTS
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Since sum{3^(-k), k = 0,1,2,...} converges, the convergents of [1, 1/3, 1/9, 1/27, ... ] 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 = 1.119... = [1, 8, 2, 1, 242, 8, 1, 6560, 26, 1, 177146, 80, 1,...];
v = 3.668... = [3, 1, 2, 80, 1, 8, 2186, 1, 26, 59048, 1, 80, ...].
In both cases, every term of the continued fraction has the form 3^m - 1.
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MATHEMATICA
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$MaxExtraPrecision = Infinity; z = 500; t = Table[3^(-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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