Śleszyński–Pringsheim theorem: Difference between revisions

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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 $n=1,2,3,\ldots$ are real numbers and $|b_{n}|\geq |a_{n}|+1$ for all $n$ , then

{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+\ddots }}}}}}

converges absolutely to a number $f$ satisfying $0<|f|<1$ ,^[4] meaning that the series

f=\sum _{n}\left\{{\frac {A_{n}}{B_{n}}}-{\frac {A_{n-1}}{B_{n-1}}}\right\},

where $A_{n}/B_{n}$ 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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[1] Слешинскій, И. В. (1889). "Дополненiе къ замѣткѣ о сходимости непрерывныхъ дробей". Матем. Сб. (in Russian). 14 (3): 436–438.

[2] Pringsheim, A. (1898). "Ueber die Convergenz unendlicher Kettenbrüche". Münch. Ber. (in German). 28: 295–324. JFM 29.0178.02.

[3] 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.

[4] Lorentzen, L.; Waadeland, H. (2008). Continued Fractions: Convergence theory. Atlantic Press. p. 129.

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+{{Short description|Criterion for convergence of continued fractions}}
-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&uunl;nch. Ber.|volume=28|pages=295&ndash;324|year=1898}}</ref> in the late 19th century.<ref>W.J.Thron ({{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}}) provides evidence that Pringsheim was aware of the work of Śleszyński before he published his article.</ref>
+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 Kettenbrüche|language=German|journal=Münch. 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
+It states that if <math>a_n</math>, <math>b_n</math>, for <math>n=1,2,3,\ldots</math> are [[real number]]s and <math>|b_n|\geq |a_n|+1</math> for all <math>n</math>, then
-: <math> \cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+ \cdots}}} </math>
+: <math> \cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+ \ddots}}} </math>
-[[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>
+converges absolutely to a number <math>f</math> satisfying <math>0<|f|<1</math>,<ref>{{cite book|first1=L.|last1=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 <math>A_n / B_n</math> are the [[convergent (continued fraction)|convergents]] of the continued fraction, [[absolute convergence|converges absolutely]].
 ==See also==
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 {{reflist}}
+{{DEFAULTSORT:Sleszynski-Pringsheim Theorem}}
 [[Category:Continued fractions]]
-[[Category:Mathematical theorems]]
+[[Category:Theorems in real analysis]]
+{{mathanalysis-stub}}