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 infinite continued fraction

${\displaystyle x=b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+{\cfrac {a_{4}}{b_{4}+\ddots }}}}}}}}.\,}$

This convergence problem is inherently more difficult than the corresponding problem for infinite series.