@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:-HXC2CCN6-1
  dc:modified "2024-10-18"^^xsd:date ;
  skos:prefLabel "constante de Cahen"@fr, "Cahen's constant"@en ;
  skos:broader psr:-L3LNPG9M-Q, psr:-RBFVN7DN-2 ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Constante_de_Cahen>, <https://en.wikipedia.org/wiki/Cahen%27s_constant> ;
  skos:definition """In mathematics, <b>Cahen's constant</b> is defined as the value of an infinite series of unit fractions with alternating signs:   <dl><dd><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 C=\\\\sum _{i=0}^{\\\\infty }{\\rac {(-1)^{i}}{s_{i}-1}}={\\rac {1}{1}}-{\\rac {1}{2}}+{\\rac {1}{6}}-{\\rac {1}{42}}+{\\rac {1}{1806}}-\\\\cdots \\\\approx 0.643410546288...}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>C</mi>         <mo>=</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>             <mo>=</mo>             <mn>0</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo stretchy="false">(</mo>               <mo>−<!-- − --></mo>               <mn>1</mn>               <msup>                 <mo stretchy="false">)</mo>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>i</mi>                 </mrow>               </msup>             </mrow>             <mrow>               <msub>                 <mi>s</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>i</mi>                 </mrow>               </msub>               <mo>−<!-- − --></mo>               <mn>1</mn>             </mrow>           </mfrac>         </mrow>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>1</mn>           </mfrac>         </mrow>         <mo>−<!-- − --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>2</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>6</mn>           </mfrac>         </mrow>         <mo>−<!-- − --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>42</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>1806</mn>           </mfrac>         </mrow>         <mo>−<!-- − --></mo>         <mo>⋯<!-- ⋯ --></mo>         <mo>≈<!-- ≈ --></mo>         <mn>0.643410546288...</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle C=\\\\sum _{i=0}^{\\\\infty }{\\rac {(-1)^{i}}{s_{i}-1}}={\\rac {1}{1}}-{\\rac {1}{2}}+{\\rac {1}{6}}-{\\rac {1}{42}}+{\\rac {1}{1806}}-\\\\cdots \\\\approx 0.643410546288...}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0d19f743f981dba45125861b9002bec2d825bf" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:70.8ex; height:7.009ex;" alt="{\\\\displaystyle C=\\\\sum _{i=0}^{\\\\infty }{\\rac {(-1)^{i}}{s_{i}-1}}={\\rac {1}{1}}-{\\rac {1}{2}}+{\\rac {1}{6}}-{\\rac {1}{42}}+{\\rac {1}{1806}}-\\\\cdots \\\\approx 0.643410546288...}"></span> (sequence A118227 in the OEIS)</dd></dl> Here <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_{i})_{i\\\\geq 0}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo stretchy="false">(</mo>         <msub>           <mi>s</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>           </mrow>         </msub>         <msub>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>             <mo>≥<!-- ≥ --></mo>             <mn>0</mn>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle (s_{i})_{i\\\\geq 0}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/885b4444b26b8dfea2da5dd06ccdc09aee02a2f4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.6ex; height:2.843ex;" alt="{\\\\displaystyle (s_{i})_{i\\\\geq 0}}"></span> denotes Sylvester's sequence, which is defined recursively by  <dl><dd><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 {\\egin{array}{l}s_{0}~~~=2;\\\\\\\\s_{i+1}=1+\\\\prod _{j=0}^{i}s_{j}{\\	ext{ for }}i\\\\geq 0.\\\\end{array}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mtable columnalign="left" rowspacing="4pt" columnspacing="1em">             <mtr>               <mtd>                 <msub>                   <mi>s</mi>                   <mrow class="MJX-TeXAtom-ORD">                     <mn>0</mn>                   </mrow>                 </msub>                 <mtext> </mtext>                 <mtext> </mtext>                 <mtext> </mtext>                 <mo>=</mo>                 <mn>2</mn>                 <mo>;</mo>               </mtd>             </mtr>             <mtr>               <mtd>                 <msub>                   <mi>s</mi>                   <mrow class="MJX-TeXAtom-ORD">                     <mi>i</mi>                     <mo>+</mo>                     <mn>1</mn>                   </mrow>                 </msub>                 <mo>=</mo>                 <mn>1</mn>                 <mo>+</mo>                 <munderover>                   <mo>∏<!-- ∏ --></mo>                   <mrow class="MJX-TeXAtom-ORD">                     <mi>j</mi>                     <mo>=</mo>                     <mn>0</mn>                   </mrow>                   <mrow class="MJX-TeXAtom-ORD">                     <mi>i</mi>                   </mrow>                 </munderover>                 <msub>                   <mi>s</mi>                   <mrow class="MJX-TeXAtom-ORD">                     <mi>j</mi>                   </mrow>                 </msub>                 <mrow class="MJX-TeXAtom-ORD">                   <mtext> for </mtext>                 </mrow>                 <mi>i</mi>                 <mo>≥<!-- ≥ --></mo>                 <mn>0.</mn>               </mtd>             </mtr>           </mtable>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\egin{array}{l}s_{0}~~~=2;\\\\\\\\s_{i+1}=1+\\\\prod _{j=0}^{i}s_{j}{\\	ext{ for }}i\\\\geq 0.\\\\end{array}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc1466297d59fa16b196b37b25f4ad2a1b01bdc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.092ex; height:6.843ex;" alt="{\\\\displaystyle {\\egin{array}{l}s_{0}~~~=2;\\\\\\\\s_{i+1}=1+\\\\prod _{j=0}^{i}s_{j}{\\	ext{ for }}i\\\\geq 0.\\\\end{array}}}"></span></dd></dl> Combining these fractions in pairs leads to an alternative expansion of Cahen's constant as a series of positive unit fractions formed from the terms in even positions of Sylvester's sequence. This series for Cahen's constant forms its greedy Egyptian expansion:  <dl><dd><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 C=\\\\sum {\\rac {1}{s_{2i}}}={\\rac {1}{2}}+{\\rac {1}{7}}+{\\rac {1}{1807}}+{\\rac {1}{10650056950807}}+\\\\cdots }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>C</mi>         <mo>=</mo>         <mo>∑<!-- ∑ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <msub>               <mi>s</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>                 <mi>i</mi>               </mrow>             </msub>           </mfrac>         </mrow>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>2</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>7</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>1807</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>10650056950807</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mo>⋯<!-- ⋯ --></mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle C=\\\\sum {\\rac {1}{s_{2i}}}={\\rac {1}{2}}+{\\rac {1}{7}}+{\\rac {1}{1807}}+{\\rac {1}{10650056950807}}+\\\\cdots }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc06a2318c69c1d90b826ed1fd3aaa3aa837fa0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:55.932ex; height:5.509ex;" alt="C=\\\\sum {\\rac {1}{s_{2i}}}={\\rac {1}{2}}+{\\rac {1}{7}}+{\\rac {1}{1807}}+{\\rac {1}{10650056950807}}+\\\\cdots "></span></dd></dl> This constant is named after Eugène Cahen (also known for the Cahen–Mellin integral), who was the first to introduce it and prove its irrationality. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Cahen%27s_constant">https://en.wikipedia.org/wiki/Cahen%27s_constant</a>)"""@en, """En mathématiques, la <b>constante de Cahen</b> est définie comme une somme infinie de fractions unitaires, avec des signes alternés, à partir de la suite de Sylvester <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_{i})}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo stretchy="false">(</mo>         <msub>           <mi>s</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>           </mrow>         </msub>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle (s_{i})}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5daae490db972bf983a32e16090c8e35a6c9b21d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.699ex; height:2.843ex;" alt="(s_{i})"></span> :  <dl><dd><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 C=\\\\sum _{i=0}^{\\\\infty }{\\rac {(-1)^{i}}{s_{i}-1}}={\\rac {1}{1}}-{\\rac {1}{2}}+{\\rac {1}{6}}-{\\rac {1}{42}}+{\\rac {1}{1806}}-\\\\cdots \\\\approx 0{,}64341054629}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>C</mi>         <mo>=</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>             <mo>=</mo>             <mn>0</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo stretchy="false">(</mo>               <mo>−<!-- − --></mo>               <mn>1</mn>               <msup>                 <mo stretchy="false">)</mo>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>i</mi>                 </mrow>               </msup>             </mrow>             <mrow>               <msub>                 <mi>s</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>i</mi>                 </mrow>               </msub>               <mo>−<!-- − --></mo>               <mn>1</mn>             </mrow>           </mfrac>         </mrow>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>1</mn>           </mfrac>         </mrow>         <mo>−<!-- − --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>2</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>6</mn>           </mfrac>         </mrow>         <mo>−<!-- − --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>42</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>1806</mn>           </mfrac>         </mrow>         <mo>−<!-- − --></mo>         <mo>⋯<!-- ⋯ --></mo>         <mo>≈<!-- ≈ --></mo>         <mn>0,643</mn>         <mn>41054629</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle C=\\\\sum _{i=0}^{\\\\infty }{\\rac {(-1)^{i}}{s_{i}-1}}={\\rac {1}{1}}-{\\rac {1}{2}}+{\\rac {1}{6}}-{\\rac {1}{42}}+{\\rac {1}{1806}}-\\\\cdots \\\\approx 0{,}64341054629}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b5d964124c87f3ada16ede54091a41f0c06214" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:67.697ex; height:7.009ex;" alt="{\\\\displaystyle C=\\\\sum _{i=0}^{\\\\infty }{\\rac {(-1)^{i}}{s_{i}-1}}={\\rac {1}{1}}-{\\rac {1}{2}}+{\\rac {1}{6}}-{\\rac {1}{42}}+{\\rac {1}{1806}}-\\\\cdots \\\\approx 0{,}64341054629}"></span>.</dd></dl> En regroupant ces fractions deux par deux, on peut aussi voir cette constante comme la somme des inverses des termes d'indices pairs de la suite de Sylvester ; cette représentation de la constante de Cahen est son développement par l'algorithme glouton pour les fractions égyptiennes :  <dl><dd><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 C=\\\\sum _{j=0}^{\\\\infty }{\\rac {1}{s_{2j}}}={\\rac {1}{2}}+{\\rac {1}{7}}+{\\rac {1}{1807}}+{\\rac {1}{10650056950807}}+\\\\cdots }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>C</mi>         <mo>=</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>j</mi>             <mo>=</mo>             <mn>0</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <msub>               <mi>s</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>                 <mi>j</mi>               </mrow>             </msub>           </mfrac>         </mrow>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>2</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>7</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>1807</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>10650056950807</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mo>⋯<!-- ⋯ --></mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle C=\\\\sum _{j=0}^{\\\\infty }{\\rac {1}{s_{2j}}}={\\rac {1}{2}}+{\\rac {1}{7}}+{\\rac {1}{1807}}+{\\rac {1}{10650056950807}}+\\\\cdots }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be89137382156d560902a1a0a7d0a31c9f1f2a9b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:56.042ex; height:7.176ex;" alt="{\\\\displaystyle C=\\\\sum _{j=0}^{\\\\infty }{\\rac {1}{s_{2j}}}={\\rac {1}{2}}+{\\rac {1}{7}}+{\\rac {1}{1807}}+{\\rac {1}{10650056950807}}+\\\\cdots }"></span>.</dd></dl> Son nom vient d'Eugène Cahen, qui est le premier à l'avoir formulée et étudiée 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Constante_de_Cahen">https://fr.wikipedia.org/wiki/Constante_de_Cahen</a>)"""@fr ;
  dc:created "2023-08-03"^^xsd:date .

psr:-RBFVN7DN-2
  skos:prefLabel "mathematical constant"@en, "constante mathématique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-HXC2CCN6-1 .

psr:-L3LNPG9M-Q
  skos:prefLabel "nombre transcendant"@fr, "transcendental number"@en ;
  a skos:Concept ;
  skos:narrower psr:-HXC2CCN6-1 .