@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-C7167V5J-J
  skos:broader psr:-B3GGSQMX-3 ;
  skos:prefLabel "Cauchy condensation test"@en, "test de condensation de Cauchy"@fr ;
  dc:created "2023-08-03"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Test_de_condensation_de_Cauchy>, <https://en.wikipedia.org/wiki/Cauchy_condensation_test> ;
  skos:definition """En analyse mathématique, le <b>test de condensation de Cauchy</b>, démontré par Augustin Louis Cauchy, est un critère de convergence pour les séries&nbsp;: pour toute suite réelle positive décroissante <span class="texhtml">(<i>a<sub>n</sub></i>)</span>, on a
<br/>
<br/><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle S:=\\\\sum _{n\\\\geq 1}a_{n}<+\\\\infty {\\	ext{ si et seulement si }}T:=\\\\sum _{k\\\\geq 0}2^{k}a_{2^{k}}<+\\\\infty }">
<br/>  <semantics>
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<br/>        <mi>S</mi>
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<br/>          <mo>∑<!-- ∑ --></mo>
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<br/>            <mi>n</mi>
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<br/>          <mi>a</mi>
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<br/>          <mtext>&nbsp;si et seulement si&nbsp;</mtext>
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<br/>          <mo>∑<!-- ∑ --></mo>
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle S:=\\\\sum _{n\\\\geq 1}a_{n}&lt;+\\\\infty {\\	ext{ si et seulement si }}T:=\\\\sum _{k\\\\geq 0}2^{k}a_{2^{k}}&lt;+\\\\infty }</annotation>
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<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111bfea7b5a10098fd0528890aa19a2050bf7e84" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:58.466ex; height:5.843ex;" alt="{\\\\displaystyle S:=\\\\sum _{n\\\\geq 1}a_{n}<+\\\\infty {\\	ext{ si et seulement si }}T:=\\\\sum _{k\\\\geq 0}2^{k}a_{2^{k}}<+\\\\infty }"></span></center>
<br/>et plus précisément
<br/>
<br/><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle S\\\\leq T\\\\leq 2S}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>S</mi>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <mi>T</mi>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <mn>2</mn>
<br/>        <mi>S</mi>
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle S\\\\leq T\\\\leq 2S}</annotation>
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<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4be319acee022741bee4a821464095938145279" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.994ex; height:2.343ex;" alt="{\\\\displaystyle S\\\\leq T\\\\leq 2S}"></span>.</center> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Test_de_condensation_de_Cauchy">https://fr.wikipedia.org/wiki/Test_de_condensation_de_Cauchy</a>)"""@fr, """In mathematics, the <b>Cauchy condensation test</b>, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f(n)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>n</mi>
<br/>        <mo stretchy="false">)</mo>
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<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(n)}</annotation>
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<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="f(n)"></span> of non-negative real numbers, the series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\	extstyle \\\\sum \\\\limits _{n=1}^{\\\\infty }f(n)}">
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<br/>            <mi>n</mi>
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<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
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<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>n</mi>
<br/>        <mo stretchy="false">)</mo>
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<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\	extstyle \\\\sum \\\\limits _{n=1}^{\\\\infty }f(n)}</annotation>
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<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18dd83b3ab2d1a77353e06d868e2086300aaacf7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:7.956ex; height:6.009ex;" alt="{\\	extstyle \\\\sum \\\\limits _{n=1}^{\\\\infty }f(n)}"></span> converges if and only if the "condensed" series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\	extstyle \\\\sum \\\\limits _{n=0}^{\\\\infty }2^{n}f(2^{n})}">
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<br/>            <mi>n</mi>
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<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
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<br/>          <mn>2</mn>
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<br/>            <mi>n</mi>
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<br/>            <mi>n</mi>
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<br/>    <annotation encoding="application/x-tex">{\\	extstyle \\\\sum \\\\limits _{n=0}^{\\\\infty }2^{n}f(2^{n})}</annotation>
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<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84cf06ea441f00ac60218d6d3f3c8c676fef33aa" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:11.324ex; height:6.176ex;" alt="{\\	extstyle \\\\sum \\\\limits _{n=0}^{\\\\infty }2^{n}f(2^{n})}"></span> converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Cauchy_condensation_test">https://en.wikipedia.org/wiki/Cauchy_condensation_test</a>)"""@en ;
  dc:modified "2023-08-03"^^xsd:date ;
  skos:inScheme psr: ;
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psr:-B3GGSQMX-3
  skos:prefLabel "série"@fr, "series"@en ;
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  skos:narrower psr:-C7167V5J-J .