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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>
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


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
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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: <math> \cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+ \cdots}}} </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
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:<math> f = \sum_n \left\{ \frac{A_n}{B_n} - \frac{A_{n-1}}{B_{n-1}}\right\},</math>
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


where ''A''<sub>''n''</sub>&nbsp;/&nbsp;''B''<sub>''n''</sub> are the [[convergent (continued fraction)|convergents]] of the continued fraction, converges absolutely.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


==See also==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
*[[Convergence problem]]


== Notes and references ==
==Demos==
{{reflist}}


{{DEFAULTSORT:Sleszynski–Pringsheim Theorem}}
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
[[Category:Continued fractions]]
 
[[Category:Theorems in real analysis]]
 
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
 
==Test pages ==
 
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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