A229986 - OEIS

A229986

Decimal expansion of the upper limit of the convergents of the continued fraction [1, 1/3, 1/9, 1/27, ... ].

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Since sum{3^(-k), k = 0,1,2,...} converges, the convergents of [1, 1/3, 1/9, 1/27, ... ] 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..86.

EXAMPLE

u = 1.119... = [1, 8, 2, 1, 242, 8, 1, 6560, 26, 1, 177146, 80, 1,...];

v = 3.668... = [3, 1, 2, 80, 1, 8, 2186, 1, 26, 59048, 1, 80, ...].

In both cases, every term of the continued fraction has the form 3^m - 1.

MATHEMATICA

$MaxExtraPrecision = Infinity; z = 500; t = Table[3^(-n), {n, 0, z}]; u = N[Convergents[t][[z - 1]], 120]; v = N[Convergents[t][[z]], 120];

RealDigits[u] (* A229985 *)

RealDigits[v] (* A229986 *)

CROSSREFS

Cf. A229985, A024023.