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.
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
CROSSREFS
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