@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:-R2ZQC914-N
  skos:prefLabel "suite"@fr, "sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-K39D3VZK-8 .

psr:-K39D3VZK-8
  skos:prefLabel "produit infini"@fr, "infinite product"@en ;
  dc:created "2023-08-21"^^xsd:date ;
  skos:broader psr:-R2ZQC914-N ;
  skos:definition """In mathematics, for a sequence of complex numbers <i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, <i>a</i><sub>3</sub>, ... the <b>infinite product</b>
<br/>
<br/><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 \\\\prod _{n=1}^{\\\\infty }a_{n}=a_{1}a_{2}a_{3}\\\\cdots }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <munderover>
<br/>          <mo>∏<!-- ∏ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>3</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>⋯<!-- ⋯ --></mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\prod _{n=1}^{\\\\infty }a_{n}=a_{1}a_{2}a_{3}\\\\cdots }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac819110195d1950faf5b6fde51db859012a8370" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:18.983ex; height:6.843ex;" alt="{\\\\displaystyle \\\\prod _{n=1}^{\\\\infty }a_{n}=a_{1}a_{2}a_{3}\\\\cdots }"></span></dd></dl>
<br/>is defined to be the limit of the partial products <i>a</i><sub>1</sub><i>a</i><sub>2</sub>...<i>a</i><sub><i>n</i></sub> as <i>n</i> increases without bound. The product is said to <i>converge</i> when the limit exists and is not zero. Otherwise the product is said to <i>diverge</i>.  A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence <i>a</i><sub><i>n</i></sub> as <i>n</i> increases without bound must be 1, while the converse is in general not true.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Infinite_product">https://en.wikipedia.org/wiki/Infinite_product</a>)"""@en, """En mathématiques, étant donné une suite de nombres complexes <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 (a_{n})_{n\\\\in \\\\mathbb {N} }}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>∈<!-- ∈ --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="double-struck">N</mi>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (a_{n})_{n\\\\in \\\\mathbb {N} }}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f0c27000fbc03d920b5b678949a0043ad269bd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.759ex; height:2.843ex;" alt="(a_{n})_{{n\\\\in \\\\mathbb{N} }}"></span>, on définit le <b>produit infini</b> de la suite comme la limite, si elle existe, des produits partiels <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 a_{0}a_{1}\\\\dots a_{N}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>…<!-- … --></mo>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>N</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle a_{0}a_{1}\\\\dots a_{N}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b7cf99dcf4d514ef7a3542f36b6a396bdf0856" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.987ex; height:2.009ex;" alt="{\\\\displaystyle a_{0}a_{1}\\\\dots a_{N}}"></span> quand <span class="texhtml mvar" style="font-style:italic;">N</span> tend vers l'infini&nbsp;; 
<br/>De même qu'une série utilise la lettre <span class="texhtml">Σ</span>, un produit infini utilise la lettre grecque <span class="texhtml">Π</span> (pi majuscule)&nbsp;:
<br/>
<br/><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 a_{0}\\\\cdot a_{1}\\\\cdot a_{2}\\\\dots =\\\\prod _{n=0}^{\\\\infty }a_{n}{\\\\overset {\\	ext{def}}{=}}\\\\lim _{N\\	o \\\\infty }\\\\displaystyle \\\\prod _{n=0}^{N}a_{n}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>⋯<!-- ⋯ --></mo>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∏<!-- ∏ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mi>a</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mover>
<br/>            <mo>=</mo>
<br/>            <mtext>def</mtext>
<br/>          </mover>
<br/>        </mrow>
<br/>        <munder>
<br/>          <mo movablelimits="true" form="prefix">lim</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>N</mi>
<br/>            <mo stretchy="false">→<!-- → --></mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munder>
<br/>        <mstyle displaystyle="true" scriptlevel="0">
<br/>          <munderover>
<br/>            <mo>∏<!-- ∏ --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>n</mi>
<br/>              <mo>=</mo>
<br/>              <mn>0</mn>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>N</mi>
<br/>            </mrow>
<br/>          </munderover>
<br/>          <msub>
<br/>            <mi>a</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>n</mi>
<br/>            </mrow>
<br/>          </msub>
<br/>        </mstyle>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle a_{0}\\\\cdot a_{1}\\\\cdot a_{2}\\\\dots =\\\\prod _{n=0}^{\\\\infty }a_{n}{\\\\overset {\\	ext{def}}{=}}\\\\lim _{N\\	o \\\\infty }\\\\displaystyle \\\\prod _{n=0}^{N}a_{n}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45d70d27ec4a8e5ad0df0b1240d83b8a7e1cbdb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.93ex; height:7.343ex;" alt="{\\\\displaystyle a_{0}\\\\cdot a_{1}\\\\cdot a_{2}\\\\dots =\\\\prod _{n=0}^{\\\\infty }a_{n}{\\\\overset {\\	ext{def}}{=}}\\\\lim _{N\\	o \\\\infty }\\\\displaystyle \\\\prod _{n=0}^{N}a_{n}}"></span>.</dd> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Produit_infini">https://fr.wikipedia.org/wiki/Produit_infini</a>)"""@fr ;
  a skos:Concept ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Infinite_product>, <https://fr.wikipedia.org/wiki/Produit_infini> ;
  skos:inScheme psr: ;
  dc:modified "2023-08-21"^^xsd:date .