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 infinite continued fraction x = b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 + ⋱ . {\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.