A229981 - OEIS

%I #7 Oct 09 2013 12:05:07

%S 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,

%T 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,

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

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

%C 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).

%e u = 1.28507295... = [1, 3, 1, 1, 31, 3, 1, 255, 7, 1, 2047,...];

%e v = 2.51538415... = [2, 1, 1, 15, 1, 3, 127, 1, 7, 1023, 1, 15,...].

%e In both cases, every term of the continued fraction has the form 2^m - 1.

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

%t RealDigits[u] (* A229981 *)

%t RealDigits[v] (* A229982 *)

%Y Cf. A229982.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Oct 06 2013