

A229981


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


2



1, 2, 8, 5, 0, 7, 2, 9, 5, 6, 6, 6, 2, 4, 3, 1, 9, 8, 3, 2, 0, 3, 9, 2, 2, 7, 0, 6, 5, 1, 7, 9, 7, 1, 4, 3, 8, 8, 1, 4, 4, 0, 1, 5, 4, 6, 4, 7, 7, 9, 0, 6, 6, 6, 1, 2, 5, 9, 6, 2, 0, 5, 2, 7, 7, 9, 6, 0, 7, 4, 2, 4, 5, 8, 3, 1, 0, 9, 2, 1, 3, 6, 5, 4, 5, 9
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Since sum{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; it appears that v = 1/(u1).


LINKS

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


EXAMPLE

u = 1.28507295... = [1, 3, 1, 1, 31, 3, 1, 255, 7, 1, 2047,...];
v = 2.51538415... = [2, 1, 1, 15, 1, 3, 127, 1, 7, 1023, 1, 15,...].
In both cases, every term of the continued fraction has the form 2^m  1.


MATHEMATICA

$MaxExtraPrecision = Infinity; z = 600; t = Table[2^(n), {n, 0, z}]; u = N[Convergents[t][[z  1]], 120]; v = N[Convergents[t][[z]], 120];
RealDigits[u] (* A229981 *)
RealDigits[v] (* A229982 *)


CROSSREFS

Cf. A229982.
Sequence in context: A054671 A203269 A011058 * A188731 A188617 A154157
Adjacent sequences: A229978 A229979 A229980 * A229982 A229983 A229984


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Oct 06 2013


STATUS

approved



