A229982 Decimal expansion of the upper limit of the convergents of the continued fraction [1, 1/2, 1/4, 1/8, ... ].

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

Offset

1,1

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.

Links

Table of n, a(n) for n=1..105.

Example

u = 1.2850729... = [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. A229981.

Sequence in context: A163331 A011034 A191332 * A327123 A289848 A258020

Adjacent sequences: A229979 A229980 A229981 * A229983 A229984 A229985

Keyword

nonn,cons

Author

Clark Kimberling, Oct 06 2013

Extensions

More terms from and example corrected by Rick L. Shepherd, Jan 10 2014

Status

approved