Sum[((35 - 88 k + 56 k^2) (3 k)! (-2 + 2 k)!^6)/ ((-1)^k (16 (-1 + k)^5 (-1 + k)!^2 k! (-3 + 2 k)!^3 (-2 + 6 k)!)), {k, 2, Infinity}] == (10 - Pi^2)/4

Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["(", RowBox[List["35", "-", RowBox[List["88", " ", "k"]], "+", RowBox[List["56", " ", SuperscriptBox["k", "2"]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "k"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], ")"]], "6"]]], RowBox[List["16", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "k"]], ")"]], "5"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "k"]], ")"]], "!"]], ")"]], "2"], " ", RowBox[List["k", "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], ")"]], "3"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["6", " ", "k"]]]], ")"]], "!"]]]]]]], "\[Equal]", FractionBox[RowBox[List["10", "-", SuperscriptBox["\[Pi]", "2"]]], "4"]]]]]

<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> 2 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 35 </mn> <mo> - </mo> <mrow> <mn> 88 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mn> 56 </mn> <mo> &#8290; </mo> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> </mrow> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mn> 5 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mn> 10 </mn> <mo> - </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 35 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 88 </cn> <ci> k </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 56 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <factorial /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <cn type='integer'> -2 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <cn type='integer'> -3 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <factorial /> <apply> <plus /> <cn type='integer'> -2 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 10 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "2"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "k_"]]], " ", RowBox[List["(", RowBox[List["35", "-", RowBox[List["88", " ", "k_"]], "+", RowBox[List["56", " ", SuperscriptBox["k_", "2"]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "k_"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "k_"]]]], ")"]], "!"]], ")"]], "6"]]], RowBox[List["16", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "k_"]], ")"]], "5"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "k_"]], ")"]], "!"]], ")"]], "2"], " ", RowBox[List["k_", "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "k_"]]]], ")"]], "!"]], ")"]], "3"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["6", " ", "k_"]]]], ")"]], "!"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["10", "-", SuperscriptBox["\[Pi]", "2"]]], ")"]]]]]]]]

Troy Kessler

2007-05-02