

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
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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 oddnumbered convergents converge, as do the evennumbered 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 A298063
Adjacent sequences: A229984 A229985 A229986 * A229988 A229989 A229990


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Oct 06 2013


STATUS

approved



