Sum[k^3/(2^k Binomial[3 k, k]), {k, 1, Infinity}] == 20517/78125 + (21331 Pi)/781250 + (774 Log[2])/390625

Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", SuperscriptBox["k", "3"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["3", " ", "k"]], ",", "k"]], "]"]]]]], "\[Equal]", RowBox[List[FractionBox["20517", "78125"], "+", FractionBox[RowBox[List["21331", " ", "\[Pi]"]], "781250"], "+", FractionBox[RowBox[List["774", " ", RowBox[List["Log", "[", "2", "]"]]]], "390625"]]]]]]]

<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> <msup> <mi> k </mi> <mn> 3 </mn> </msup> </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> 20517 </mn> <mn> 78125 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 21331 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 781250 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 774 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mn> 390625 </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> <apply> <power /> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> <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'> 20517 <sep /> 78125 </cn> <apply> <times /> <cn type='integer'> 21331 </cn> <pi /> <apply> <power /> <cn type='integer'> 781250 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 774 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 390625 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k_"]]], " ", SuperscriptBox["k_", "3"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["3", " ", "k_"]], ",", "k_"]], "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["20517", "78125"], "+", FractionBox[RowBox[List["21331", " ", "\[Pi]"]], "781250"], "+", FractionBox[RowBox[List["774", " ", RowBox[List["Log", "[", "2", "]"]]]], "390625"]]]]]]]

2002-12-18