Convergence Problem - Van Vleck's Theorem

Van Vleck's Theorem

Jones and Thron attribute the following result to Van Vleck. Suppose that all the a_i are equal to 1, and all the b_i have arguments with:

$- \pi /2 + \epsilon < \arg ( b_i) < \pi / 2 - \epsilon, i \geq 1,$

with epsilon being any positive number less than . In other words, all the b_i are inside a wedge which has its vertex at the origin, has an opening angle of, and is symmetric around the positive real axis. Then f_i, the ith convergent to the continued fraction, is finite and has an argument:

$- \pi /2 + \epsilon < \arg ( f_i ) < \pi / 2 - \epsilon, i \geq 1.$

Also, the sequence of even convergents will converge, as will the sequence of odd convergents. The continued fraction itself will converge if and only if the sum of all the |b_i| diverges.

Read more about this topic: Convergence Problem

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