OFFSET
1,2
COMMENTS
Since sum{(-1)*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 = 1.401284... = [1, 2, 2, 30, 1, 2, 1, 254, 1, 6, 1, 2046, 1, ...];
v = -0.48360... = [0, -2, -14, -1, -2, -1, -126, -1, -6, -1, ...].
Every term of the continued fraction of u is 1 or has the form -2 + 2^m; every term for v is -1 or has the form 2 - 2^m.
MATHEMATICA
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 06 2013
STATUS
approved