Pi^4 == (27/164) Sum[(1/2^(12 k)) (2048/(1 + 24 k)^4 - 38912/(2 + 24 k)^4 + 81920/(3 + 24 k)^4 - 2048/(4 + 24 k)^4 - 512/(5 + 24 k)^4 - 23552/(6 + 24 k)^4 + 256/(7 + 24 k)^4 - 27648/(8 + 24 k)^4 - 10240/(9 + 24 k)^4 - 2432/(10 + 24 k)^4 - 64/(11 + 24 k)^4 - 3584/(12 + 24 k)^4 - 32/(13 + 24 k)^4 - 608/(14 + 24 k)^4 - 1280/(15 + 24 k)^4 - 1728/(16 + 24 k)^4 + 8/(17 + 24 k)^4 - 368/(18 + 24 k)^4 - 4/(19 + 24 k)^4 - 8/(20 + 24 k)^4 + 160/(21 + 24 k)^4 - 38/(22 + 24 k)^4 + 1/(23 + 24 k)^4), {k, 0, Infinity}]

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</mo> <mn> 17 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 368 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 18 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 4 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 19 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 8 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 20 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 160 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 21 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 38 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 22 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 23 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 2048 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='rational'> 27 <sep /> 164 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> 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<power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 512 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 23552 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <apply> 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</cn> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 13 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 608 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 14 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1280 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 15 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> 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<apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 19 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 20 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 160 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 21 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 38 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 22 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 23 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2048 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 24 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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G.Huvent (2006)

2007-05-02