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 A229987 Decimal expansion of the upper limit of the convergents of the continued fraction [1, -1/2, 1/4, -1/8, ... ]. 2
 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, 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, 0, 5, 2, 6, 8, 1, 6, 8, 3, 3, 2, 4, 0, 1, 5, 1, 1, 3 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS EXAMPLE 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. MATHEMATICA \$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 *) RealDigits[v] (* A229988 *) CROSSREFS Cf. A229988. Sequence in context: A198637 A179296 A196817 * A096793 A155998 A298063 Adjacent sequences:  A229984 A229985 A229986 * A229988 A229989 A229990 KEYWORD nonn,cons AUTHOR Clark Kimberling, Oct 06 2013 STATUS approved

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Last modified December 9 16:41 EST 2018. Contains 318023 sequences. (Running on oeis4.)