Śleszyński–Pringsheim theorem explained

In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński^[1] and Alfred Pringsheim^[2] in the late 19th century.^[3]

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 ,^[4] meaning that the seriesf = \sum _{n} \left\{ (A _{n})/(B _{n}) - (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

Notes and References

1. Слешинскій. И. В.. Дополненiе къ замѣткѣ о сходимости непрерывныхъ дробей. Матем. Сб.. 14. 3. 1889. 436 - 438. Russian.
2. 29.0178.02. Pringsheim. A.. Ueber die Convergenz unendlicher Kettenbrüche. German. Münch. Ber.. 28. 295 - 324. 1898.
3. W.J.Thron has found evidence that Pringsheim was aware of the work of Śleszyński before he published his article; see 1192192. Thron. W. J.. Should the Pringsheim criterion be renamed the Śleszyński criterion?. Comm. Anal. Theory Contin. Fractions. 1. 1992. 13 - 20.
4. Book: L.. Lorentzen. H.. Waadeland. Continued Fractions: Convergence theory. Atlantic Press. 2008. 129.