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http://functions.wolfram.com/06.16.23.0005.01
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Sum[HarmonicNumber[k]^3/2^k, {k, 1, Infinity}] ==
Zeta[3] + (1/3) Pi^2 Log[2] + Log[2]^3/3
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[SuperscriptBox[RowBox[List["HarmonicNumber", "[", "k", "]"]], "3"], SuperscriptBox["2", "k"]]]], "\[Equal]", RowBox[List[RowBox[List["Zeta", "[", "3", "]"]], "+", RowBox[List[FractionBox["1", "3"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Log", "[", "2", "]"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "3"], "3"]]]]]]]
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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> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> k </mi> <mn> 3 </mn> </msubsup> <msup> <mn> 2 </mn> <mi> k </mi> </msup> </mfrac> </mrow> <mo>  </mo> <mrow> <mfrac> <mrow> <msup> <mi> log </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mn> 3 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </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 /> <apply> <ci> HarmonicNumber </ci> <ci> k </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], FractionBox[SuperscriptBox[RowBox[List["HarmonicNumber", "[", "k_", "]"]], "3"], SuperscriptBox["2", "k_"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Zeta", "[", "3", "]"]], "+", RowBox[List[FractionBox["1", "3"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Log", "[", "2", "]"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "3"], "3"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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