A229981 Decimal expansion of the lower limit of the convergents of the continued fraction [1, 1/2, 1/4, 1/8, ... ].

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

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 odd-numbered convergents converge, as do the even-numbered convergents. In the Example section, these limits are denoted by u and v; it appears that v = 1/(u-1).

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: A203269 A011058 A368644 * A188731 A188617 A154157

Adjacent sequences: A229978 A229979 A229980 * A229982 A229983 A229984

Keyword

nonn,cons

Author

Clark Kimberling, Oct 06 2013

Status

approved