Sum[Product[1/Subscript[k, j]^2, {j, 1, m}], {Subscript[k, 1], 1, Infinity}, {Subscript[k, 2], Subscript[k, 2] + 1, Infinity}, …, {Subscript[k, m], Subscript[k, m - 1] + 1, Infinity}] == Pi^(2*m)/(2*m + 1)! /; m ∈ Integers && m > 0

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mi> m </mi> </msub> <mo> = </mo> <mrow> <msub> <mi> k </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mfrac> <mn> 1 </mn> <msubsup> <mi> k </mi> <mi> j </mi> <mn> 2 </mn> </msubsup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <msup> <mi> &#960; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> m </ci> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> j </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18