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 ${\displaystyle a_{n}}$, ${\displaystyle b_{n}}$, for ${\displaystyle n=1,2,3,\ldots }$ are real numbers and ${\displaystyle |b_{n}|\geq |a_{n}|+1}$ for all ${\displaystyle n}$, then

${\displaystyle {\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+\ddots }}}}}}}$

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

${\displaystyle f=\sum _{n}\left\{{\frac {A_{n}}{B_{n}}}-{\frac {A_{n-1}}{B_{n-1}}}\right\},}$

where ${\displaystyle A_{n}/B_{n}}$ are the convergents of the continued fraction, converges absolutely.