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 an, bn, for n = 1, 2, 3, ... are real numbers and |bn| ≥ |an| + 1 for all n, then
where An / Bn are the convergents of the continued fraction, converges absolutely.
Notes and references
- 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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