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60 | 60 | div.csl-bib-body { } |
61 | 61 | div.csl-entry { |
62 | 62 | clear: both; |
63 | | - margin-bottom: 0em; |
64 | 63 | } |
65 | 64 | .hanging-indent div.csl-entry { |
66 | 65 | margin-left:2em; |
@@ -308,7 +307,7 @@ <h2 data-number="1" data-anchor-id="introducción"><span class="header-section-n |
308 | 307 | </section> |
309 | 308 | <section id="números-tetraédricos" class="level2" data-number="2"> |
310 | 309 | <h2 data-number="2" data-anchor-id="números-tetraédricos"><span class="header-section-number">2</span> NÚMEROS TETRAÉDRICOS</h2> |
311 | | -<p>Los números tetraédricos representan el número de esferas que pueden apilarse formando un <em>tetraedro regular</em> <span class="citation" data-cites="wikiTetrahedralNumber"><a href="#ref-wikiTetrahedralNumber" role="doc-biblioref">[2]</a></span> <span class="citation" data-cites="mathworldTetrahedralNumber"><a href="#ref-mathworldTetrahedralNumber" role="doc-biblioref">[3]</a></span> y están definidos por la fórmula</p> |
| 310 | +<p>Los números tetraédricos representan el número de esferas que pueden apilarse formando un <em>tetraedro regular</em> <span class="citation" data-cites="wikiTetrahedralNumber">[<a href="#ref-wikiTetrahedralNumber" role="doc-biblioref">2</a>]</span> <span class="citation" data-cites="mathworldTetrahedralNumber">[<a href="#ref-mathworldTetrahedralNumber" role="doc-biblioref">3</a>]</span> y están definidos por la fórmula</p> |
312 | 311 | <p><span id="eq-tet001"><span class="math display"> |
313 | 312 | T_n = \frac{n(n+1)(n+2)}{6} = \binom{n+2}{3} |
314 | 313 | \tag{1}</span></span></p> |
@@ -345,7 +344,7 @@ <h2 data-number="4" data-anchor-id="descomposición-en-fracciones-parciales"><sp |
345 | 344 | </section> |
346 | 345 | <section id="números-armónicos" class="level2" data-number="5"> |
347 | 346 | <h2 data-number="5" data-anchor-id="números-armónicos"><span class="header-section-number">5</span> NÚMEROS ARMÓNICOS</h2> |
348 | | -<p>Estas tres series se pueden sumar facilmente teniendo en cuenta las siguientes identidades para los <em>números armónicos</em> <span class="math inline">H_n</span> <span class="citation" data-cites="wikiHarmonicNumber"><a href="#ref-wikiHarmonicNumber" role="doc-biblioref">[4]</a></span>:</p> |
| 347 | +<p>Estas tres series se pueden sumar facilmente teniendo en cuenta las siguientes identidades para los <em>números armónicos</em> <span class="math inline">H_n</span> <span class="citation" data-cites="wikiHarmonicNumber">[<a href="#ref-wikiHarmonicNumber" role="doc-biblioref">4</a>]</span>:</p> |
349 | 348 | <p><span id="eq-tet007"><span class="math display"> |
350 | 349 | \sum_{k=1}^{n} \frac{1}{k} = H_n |
351 | 350 | \tag{7}</span></span></p> |
@@ -396,35 +395,35 @@ <h2 data-number="7" data-anchor-id="suma-infinita"><span class="header-section-n |
396 | 395 | </section> |
397 | 396 |
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398 | 397 |
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399 | | -<div id="quarto-appendix" class="default"><section class="quarto-appendix-contents" role="doc-bibliography" id="quarto-bibliography"><h2 class="anchored quarto-appendix-heading">8 REFERENCIAS</h2><div id="refs" class="references csl-bib-body" data-entry-spacing="0" role="list"> |
| 398 | +<div id="quarto-appendix" class="default"><section class="quarto-appendix-contents" role="doc-bibliography" id="quarto-bibliography"><h2 class="anchored quarto-appendix-heading">8 REFERENCIAS</h2><div id="refs" class="references csl-bib-body" role="list"> |
400 | 399 | <div id="ref-psychedelic2009" class="csl-entry" role="listitem"> |
401 | | -<div class="csl-left-margin">[1] </div><div class="csl-right-inline">P. Geometry, <span>«TETRAHEDRAL NUMBERS RECIPROCALS SUM»</span>. 2009. Disponible en: <a href="https://psychedelicgeometry.wordpress.com/2009/12/25/tetrahedral-numbers-reciprocals-sum/">https://psychedelicgeometry.wordpress.com/2009/12/25/tetrahedral-numbers-reciprocals-sum/</a></div> |
| 400 | +<div class="csl-left-margin">[1] </div><div class="csl-right-inline"><span class="smallcaps">Geometry</span>, P. (2009). <a href="https://psychedelicgeometry.wordpress.com/2009/12/25/tetrahedral-numbers-reciprocals-sum/">TETRAHEDRAL NUMBERS RECIPROCALS SUM</a>.</div> |
402 | 401 | </div> |
403 | 402 | <div id="ref-wikiTetrahedralNumber" class="csl-entry" role="listitem"> |
404 | | -<div class="csl-left-margin">[2] </div><div class="csl-right-inline"><span>«Tetrahedral number»</span>. Wikipedia; <a href="https://en.wikipedia.org/wiki/Tetrahedral_number" class="uri">https://en.wikipedia.org/wiki/Tetrahedral_number</a>.</div> |
| 403 | +<div class="csl-left-margin">[2] </div><div class="csl-right-inline"><span class="smallcaps">Anón</span>. Tetrahedral number.</div> |
405 | 404 | </div> |
406 | 405 | <div id="ref-mathworldTetrahedralNumber" class="csl-entry" role="listitem"> |
407 | | -<div class="csl-left-margin">[3] </div><div class="csl-right-inline">E. W. Weisstein, <span>«Tetrahedral Number»</span>. From MathWorld–A Wolfram Web Resource. Disponible en: <a href="http://mathworld.wolfram.com/TetrahedralNumber.html">http://mathworld.wolfram.com/TetrahedralNumber.html</a></div> |
| 406 | +<div class="csl-left-margin">[3] </div><div class="csl-right-inline"><span class="smallcaps">Weisstein</span>, E. W. <a href="http://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>.</div> |
408 | 407 | </div> |
409 | 408 | <div id="ref-wikiHarmonicNumber" class="csl-entry" role="listitem"> |
410 | | -<div class="csl-left-margin">[4] </div><div class="csl-right-inline"><span>«Harmonic number»</span>. Wikipedia; <a href="https://en.wikipedia.org/wiki/Harmonic_number" class="uri">https://en.wikipedia.org/wiki/Harmonic_number</a>.</div> |
| 409 | +<div class="csl-left-margin">[4] </div><div class="csl-right-inline"><span class="smallcaps">Anón</span>. Harmonic number.</div> |
411 | 410 | </div> |
412 | 411 | <div id="ref-oeisA000292" class="csl-entry" role="listitem"> |
413 | | -<div class="csl-left-margin">[5] </div><div class="csl-right-inline">Online Encyclopedia of Integer Sequences, <span>«Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.»</span> <a href="https://oeis.org/A000292" class="uri">https://oeis.org/A000292</a>, 2009.</div> |
| 412 | +<div class="csl-left-margin">[5] </div><div class="csl-right-inline"><span class="smallcaps">Online Encyclopedia of Integer Sequences</span>. (2009). Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.</div> |
414 | 413 | </div> |
415 | 414 | <div id="ref-caglayan2015" class="csl-entry" role="listitem"> |
416 | | -<div class="csl-left-margin">[6] </div><div class="csl-right-inline">G. Caglayan, <span>«Proof Without Words: Series of Reciprocals of Tetrahedral Numbers»</span>, <em>The College Mathematics Journal</em>, vol. 46, n.º 2, p. 130, 2015, doi: <a href="https://doi.org/10.4169/college.math.j.46.2.130">10.4169/college.math.j.46.2.130</a>.</div> |
| 415 | +<div class="csl-left-margin">[6] </div><div class="csl-right-inline"><span class="smallcaps">Caglayan</span>, G. (2015). <a href="https://doi.org/10.4169/college.math.j.46.2.130">Proof Without Words: Series of Reciprocals of Tetrahedral Numbers</a>. <em>The College Mathematics Journal</em> <strong>46</strong> 130.</div> |
417 | 416 | </div> |
418 | 417 | <div id="ref-oeisA118391" class="csl-entry" role="listitem"> |
419 | | -<div class="csl-left-margin">[7] </div><div class="csl-right-inline">J. V. Post, A. Adamchuk, G. C. Greubel, H. P. Dale, y P. Carmody, <span>«A118391: Numerator of sum of reciprocals of first n tetrahedral numbers»</span>. <a href="https://oeis.org/A118391" class="uri">https://oeis.org/A118391</a>, 2006.</div> |
| 418 | +<div class="csl-left-margin">[7] </div><div class="csl-right-inline"><span class="smallcaps">Post</span>, J. V., <span class="smallcaps">Adamchuk</span>, A., <span class="smallcaps">Greubel</span>, G. C., <span class="smallcaps">Dale</span>, H. P. y <span class="smallcaps">Carmody</span>, P. (2006). A118391: Numerator of sum of reciprocals of first n tetrahedral numbers.</div> |
420 | 419 | </div> |
421 | 420 | <div id="ref-oeisA118392" class="csl-entry" role="listitem"> |
422 | | -<div class="csl-left-margin">[8] </div><div class="csl-right-inline">J. V. Post, H. P. Dale, y G. C. Greubel, <span>«A118392: Denominator of sum of reciprocals of first n tetrahedral numbers»</span>. <a href="https://oeis.org/A118392" class="uri">https://oeis.org/A118392</a>, 2006.</div> |
| 421 | +<div class="csl-left-margin">[8] </div><div class="csl-right-inline"><span class="smallcaps">Post</span>, J. V., <span class="smallcaps">Dale</span>, H. P. y <span class="smallcaps">Greubel</span>, G. C. (2006). A118392: Denominator of sum of reciprocals of first n tetrahedral numbers.</div> |
423 | 422 | </div> |
424 | 423 | </div></section><section id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes"><h2 class="anchored quarto-appendix-heading">Notas</h2> |
425 | 424 |
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426 | 425 | <ol> |
427 | | -<li id="fn1"><p>Este artículo es una <em>traducción y actualización</em> del post original titulado <em>TETRAHEDRAL NUMBERS RECIPROCALS SUM</em> publicado el <em>viernes 25 de diciembre de 2009</em> en el blog <em>Psychedelic Geometry</em> <span class="citation" data-cites="psychedelic2009"><a href="#ref-psychedelic2009" role="doc-biblioref">[1]</a></span>.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li> |
| 426 | +<li id="fn1"><p>Este artículo es una <em>traducción y actualización</em> del post original titulado <em>TETRAHEDRAL NUMBERS RECIPROCALS SUM</em> publicado el <em>viernes 25 de diciembre de 2009</em> en el blog <em>Psychedelic Geometry</em> <span class="citation" data-cites="psychedelic2009">[<a href="#ref-psychedelic2009" role="doc-biblioref">1</a>]</span>.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li> |
428 | 427 | </ol> |
429 | 428 | </section></div></main> <!-- /main --> |
430 | 429 | <!-- partials/_footer.html --> |
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