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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

Table of n, a(n) for n=1..86.

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 A127538

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 July 26 00:44 EDT 2017. Contains 289798 sequences.