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.

See also

• Convergence problem