User:DavidCBryant/Convergence problem (continued fraction)

In analysis, the convergence problem associated with the infinite continued fraction


 * $$x = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{\ddots\,}}}}$$

is the derivation of conditions on the partial numerators ai and bi sufficient to ensure that the sequence of convergents approaches a finite limit. This problem is easily solved when all the partial numerators and partial denominators are real and positive. This problem is much more difficult when the ai and bi are allowed to assume arbitrary complex values.