Śleszyński–Pringsheim theorem

In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was 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

converges absolutely to a number ƒ satisfying 0 < |ƒ| < 1, meaning that the series

where A_n / B_n are the convergents of the continued fraction, converges absolutely.