Convergence Problem

In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators a_i and partial denominators b_i 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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