Pincherle polynomials: Difference between revisions

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In mathematics, the '''Śleszyński–Pringsheim theorem''' is a statement about [[Convergent (continued fraction)|convergence]] of certain [[continued fraction]]s. It was discovered by [[Ivan Śleszyński]]<ref>{{cite journal|last=Слешинскій|first=И. В.|title=Дополненiе къ замѣткѣ о сходимости непрерывныхъ дробей|journal=Матем. сб.|volume=14|issue=3|year=1889|pages=436&ndash;438|url=http://mi.mathnet.ru/msb7210|language=Russian}}</ref> and [[Alfred Pringsheim]]<ref>{{cite journal|jfm=29.0178.02|last=Pringsheim|first=A.|title=Ueber die Convergenz unendlicher Kettenbr&uuml;che|language=German|journal=M&uuml;nch. Ber.|volume=28|pages=295&ndash;324|year=1898}}</ref> in the late 19th century.<ref>W.J.Thron has found evidence that Pringsheim was aware of the work of Śleszyński before he published his article; see {{cite journal|mr=1192192|last=Thron|first=W. J.|title=Should the Pringsheim criterion be renamed the Śleszyński criterion?|journal=Comm. Anal. Theory Contin. Fractions|volume=1|year=1992|pages=13&ndash;20}}</ref>


It states that if ''a''<sub>''n''</sub>, ''b''<sub>''n''</sub>, for&nbsp;''n''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;... are [[real number]]s and |''b''<sub>''n''</sub>|&nbsp;≥&nbsp;|''a''<sub>''n''</sub>|&nbsp;+&nbsp;1 for all&nbsp;''n'', then


: <math> \cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+ \ddots}}} </math>
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[[absolute convergence|converges absolutely]] to a number ''&fnof;'' satisfying&nbsp;0&nbsp;<&nbsp;|''&fnof;''|&nbsp;<&nbsp;1,<ref>{{cite book|first=L.|last=Lorentzen|first2=H.|last2=Waadeland|title=Continued Fractions: Convergence theory|publisher=Atlantic Press|year=2008|page=129}}</ref> meaning that the series
 
:<math> f = \sum_n \left\{ \frac{A_n}{B_n} - \frac{A_{n-1}}{B_{n-1}}\right\},</math>
 
where ''A''<sub>''n''</sub>&nbsp;/&nbsp;''B''<sub>''n''</sub> are the [[convergent (continued fraction)|convergents]] of the continued fraction, converges absolutely.
 
==See also==
*[[Convergence problem]]
 
== Notes and references ==
{{reflist}}
 
{{DEFAULTSORT:Sleszynski-Pringsheim Theorem}}
[[Category:Continued fractions]]
[[Category:Theorems in real analysis]]
 
 
{{mathanalysis-stub}}

Latest revision as of 15:55, 23 September 2014


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