Śleszyński–Pringsheim theorem

In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions, discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century.

It states that if a_n, b_n, for n = 1, 2, 3, ... are real numbers and |b_n| ≥ |a_n| + 1 for all n, then

${\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+\cdots }}}}}}$

converges absolutely to a number ƒ satisfying 0 < |ƒ| < 1.