Sequences of convergence regions for continued fractions $K(a\sb{n}/1)$

Sufficient conditions are given for convergence of continued fractions $K({a_n}/1)$ such that ${a_n} \in {E_n},n \geqslant 1$, where $\{ {E_n}\}$ is a sequence of element regions in the complex plane. The method employed makes essential use of a nested sequence of circular disks (inclusion regions), such that the $n$th disk contains the $n$th approximant of the continued fraction. This sequence can either shrink to a point, the limit point case, or to a disk, the limit circle case. Sufficient conditions are determined for convergence of the continued fraction in the limit circle case and these conditions are incorporated in the element regions ${E_n}$. The results provide new criteria for a sequence $\{ {E_n}\}$ with unbounded regions to be an admissible sequence. They also yield generalizations of certain twin-convergence regions.