A229983 Decimal expansion of the lower limit of the convergents of the continued fraction [1/2, 1/4, 1/8, ... ].

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

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 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=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 A309549

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