Sum[k/(2^k Binomial[3 k, k]), {k, 1, Infinity}] == 81/625 + (79 Pi)/3125 - (18 Log[2])/3125

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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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> k </mi> </mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;3&quot;, &quot; &quot;, &quot;k&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mfrac> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 81 </mn> <mn> 625 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 79 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 3125 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 18 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mn> 3125 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <ci> k </ci> <apply> <power /> <apply> <ci> Binomial </ci> <apply> <times /> <cn type='integer'> 3 </cn> <ci> k </ci> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='rational'> 81 <sep /> 625 </cn> <apply> <times /> <cn type='integer'> 79 </cn> <pi /> <apply> <power /> <cn type='integer'> 3125 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 3125 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18