Śleszyński–Pringsheim theorem
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Short description: Criterion for convergence of continued fractions
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 [math]\displaystyle{ a_n }[/math], [math]\displaystyle{ b_n }[/math], for [math]\displaystyle{ n=1,2,3,\ldots }[/math] are real numbers and [math]\displaystyle{ |b_n|\geq |a_n|+1 }[/math] for all [math]\displaystyle{ n }[/math], then
- [math]\displaystyle{ \cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+ \ddots}}} }[/math]
converges absolutely to a number [math]\displaystyle{ f }[/math] satisfying [math]\displaystyle{ 0\lt |f|\lt 1 }[/math],[4] meaning that the series
- [math]\displaystyle{ f = \sum_n \left\{ \frac{A_n}{B_n} - \frac{A_{n-1}}{B_{n-1}}\right\}, }[/math]
where [math]\displaystyle{ A_n / B_n }[/math] are the convergents of the continued fraction, converges absolutely.
See also
Notes and references
- ↑ Слешинскій, И. В. (1889). "Дополненiе къ замѣткѣ о сходимости непрерывныхъ дробей" (in Russian). Матем. Сб. 14 (3): 436–438. http://mi.mathnet.ru/msb7210.
- ↑ Pringsheim, A. (1898). "Ueber die Convergenz unendlicher Kettenbrüche" (in German). Münch. Ber. 28: 295–324.
- ↑ 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.
- ↑ Lorentzen, L.; Waadeland, H. (2008). Continued Fractions: Convergence theory. Atlantic Press. p. 129.
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