# Śleszyński–Pringsheim theorem

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 an, bn, for n = 1, 2, 3, ... are real numbers and |bn| ≥ |an| + 1 for all n, then {displaystyle {cfrac {a_{1}}{b_{1}+{cfrac {a_{2}}{b_{2}+{cfrac {a_{3}}{b_{3}+ddots }}}}}}} converges absolutely to a number ƒ satisfying 0 < |ƒ| < 1,[4] meaning that the series {displaystyle f=sum _{n}left{{frac {A_{n}}{B_{n}}}-{frac {A_{n-1}}{B_{n-1}}}right},} where An / Bn are the convergents of the continued fraction, converges absolutely. See also Convergence problem 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. This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it. Categories: Continued fractionsTheorems in real analysisMathematical analysis stubs

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