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 $$n=1,2,3,\ldots$$ are real numbers and $$|b_n|\geq |a_n|+1$$ for all $$n$$, then


 * $$ \cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+ \ddots}}} $$

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


 * $$ f = \sum_n \left\{ \frac{A_n}{B_n} - \frac{A_{n-1}}{B_{n-1}}\right\},$$

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