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variants of this functions
HarmonicNumber






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > HarmonicNumber[z] > Summation > Infinite summation





http://functions.wolfram.com/06.16.23.0007.01









  


  










Input Form





Sum[(HarmonicNumber[2 k] HarmonicNumber[2 k - 1])/k^2, {k, 1, Infinity}] == (17 Pi^4)/240










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["2", " ", "k"]], "]"]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], "]"]]]], SuperscriptBox["k", "2"]]]], "\[Equal]", FractionBox[RowBox[List["17", " ", SuperscriptBox["\[Pi]", "4"]]], "240"]]]]]










MathML Form







<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> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> &#63449; </mo> <mfrac> <mrow> <mn> 17 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> </mrow> <mn> 240 </mn> </mfrac> </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> <ci> HarmonicNumber </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 17 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 240 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["2", " ", "k_"]], "]"]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["2", " ", "k_"]], "-", "1"]], "]"]]]], SuperscriptBox["k_", "2"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["17", " ", SuperscriptBox["\[Pi]", "4"]]], "240"]]]]]










Contributed by





G.Huvent (2006)










Date Added to functions.wolfram.com (modification date)





2007-05-02





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