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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/06.16.23.0003.01









  


  










Input Form





Sum[HarmonicNumber[k]/(k^2 2^k), {k, 1, Infinity}] == Zeta[3] - (1/12) Pi^2 Log[2]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["HarmonicNumber", "[", "k", "]"]], RowBox[List[SuperscriptBox["k", "2"], " ", SuperscriptBox["2", "k"]]]]]], "\[Equal]", RowBox[List[RowBox[List["Zeta", "[", "3", "]"]], "-", RowBox[List[FractionBox["1", "12"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Log", "[", "2", "]"]]]]]]]]]]










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> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> k </mi> </msub> <mrow> <msup> <mi> k </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> <mo> &#63449; </mo> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;3&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 12 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> </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> <ci> HarmonicNumber </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 12 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> </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["HarmonicNumber", "[", "k_", "]"]], RowBox[List[SuperscriptBox["k_", "2"], " ", SuperscriptBox["2", "k_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Zeta", "[", "3", "]"]], "-", RowBox[List[FractionBox["1", "12"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Log", "[", "2", "]"]]]]]]]]]]










Contributed by





G.Huvent (2006)










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





2007-05-02