

A229983


Decimal expansion of the lower limit of the convergents of the continued fraction [1/2, 1/4, 1/8, ... ].


2



6, 5, 9, 8, 9, 8, 6, 7, 8, 2, 6, 1, 3, 6, 2, 3, 6, 7, 2, 2, 4, 2, 9, 8, 1, 0, 7, 2, 6, 5, 7, 1, 8, 3, 4, 7, 7, 5, 4, 4, 2, 2, 8, 9, 7, 1, 1, 4, 6, 3, 9, 9, 5, 6, 5, 1, 6, 2, 5, 6, 5, 5, 7, 0, 4, 4, 8, 0, 0, 5, 3, 5, 9, 9, 7, 1, 3, 1, 5, 7, 2, 4, 7, 2, 7, 0, 6, 1, 7, 0, 8, 0, 0, 3, 8, 1, 8, 0, 7, 7, 9, 4, 9, 8, 9
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OFFSET

0,1


COMMENTS

Since sum{2^(k), k=1,2,...} converges, the convergents of [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=0..104.


EXAMPLE

u = 0.659898678... = [0, 1, 1, 1, 15, 1, 3, 127, 1, 7, 1023, 1,...];
v = 3.507873955... = [3, 1, 1, 31, 3, 1, 255, 7, 1, 2047, 15, ...].
In both cases, every term of the continued fraction has the form 2^m  1.


MATHEMATICA

$MaxExtraPrecision = Infinity; z = 500; t = Table[2^(n), {n, 1, z}]; u = N[Convergents[t][[z  1]], 120]; v = N[Convergents[t][[z]], 120];
RealDigits[u] (* A229983 *)
RealDigits[v] (* A229984 *)


CROSSREFS

Cf. A229982.
Sequence in context: A269768 A073230 A134881 * A251859 A100884 A187892
Adjacent sequences: A229980 A229981 A229982 * A229984 A229985 A229986


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Oct 06 2013


EXTENSIONS

Offset corrected by and more terms from Rick L. Shepherd, Jan 09 2014


STATUS

approved



