# Śleszyński–Pringsheim theorem

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, ... are real numbers and |*b*_{n}| ≥ |*a*_{n}| + 1 for all *n*, then

converges absolutely to a number *ƒ* satisfying 0 < |*ƒ*| < 1,^{[4]} meaning that the series

where *A*_{n} / *B*_{n} are the convergents of the continued fraction, converges absolutely.

## See alsoEdit

## Notes and referencesEdit

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

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