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Decimal expansion of the upper limit of the convergents of the continued fraction [1, 1/2, 1/4, 1/8, ... ].
3

%I #7 Jan 14 2014 11:28:18

%S 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,

%T 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,

%U 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

%N Decimal expansion of the upper 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.

%e u = 1.2850729... = [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. A229981.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 06 2013

%E More terms from and example corrected by _Rick L. Shepherd_, Jan 10 2014