Convergence Problem

In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators ai and partial denominators bi that are sufficient to guarantee the convergence of the continued fraction

x = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots}}}}.\,

This convergence problem for continued fractions is inherently more difficult (and also more interesting) than the corresponding convergence problem for infinite series.

Read more about Convergence Problem:  Elementary Results, Śleszyński–Pringsheim Criterion, Van Vleck's Theorem

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