Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Pi






Mathematica Notation

Traditional Notation









Constants > Pi > Series representations > Generalized power series > Expansions for Pi3





http://functions.wolfram.com/02.03.06.0057.01









  


  










Input Form





Pi^3 == (1/16) Sum[((-1)^k/1024^k) (32/(1 + 4 k)^3 + 8/(2 + 4 k)^3 + 1/(3 + 4 k)^3), {k, 0, Infinity}] + (5/2) Sum[((-1)^k/64^k) (32/(1 + 12 k)^3 - 192/(2 + 12 k)^3 + 88/(3 + 12 k)^3 - 8/(5 + 12 k)^3 + 84/(6 + 12 k)^3 - 4/(7 + 12 k)^3 + 11/(9 + 12 k)^3 - 12/(10 + 12 k)^3 + 1/(11 + 12 k)^3), {k, 0, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[SuperscriptBox["\[Pi]", "3"], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "16"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", "1024", ")"]], "k"]], " ", RowBox[List["(", RowBox[List[FractionBox["32", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox["5", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", "64", ")"]], "k"]], RowBox[List["(", RowBox[List[FractionBox["32", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["192", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["88", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["84", SuperscriptBox[RowBox[List["(", RowBox[List["6", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["4", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["11", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["12", SuperscriptBox[RowBox[List["(", RowBox[List["10", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["11", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]]]], ")"]]]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msup> <mi> &#960; </mi> <mn> 3 </mn> </msup> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 16 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <msup> <mn> 1024 </mn> <mi> k </mi> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 8 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 32 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <msup> <mn> 64 </mn> <mi> k </mi> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 192 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> + </mo> <mfrac> <mn> 88 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 8 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 84 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 4 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 11 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 12 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 10 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 11 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 32 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 16 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 1024 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 5 <sep /> 2 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 64 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 192 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 88 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 84 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 7 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 11 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 9 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 10 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 11 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["\[Pi]", "3"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "16"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[FractionBox["32", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "k"]]]], ")"]], "3"]]]], ")"]]]], SuperscriptBox["1024", "k"]]]]]], "+", RowBox[List[FractionBox["5", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[FractionBox["32", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["192", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["88", SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["8", SuperscriptBox[RowBox[List["(", RowBox[List["5", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["84", SuperscriptBox[RowBox[List["(", RowBox[List["6", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["4", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["11", SuperscriptBox[RowBox[List["(", RowBox[List["9", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "-", FractionBox["12", SuperscriptBox[RowBox[List["(", RowBox[List["10", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["11", "+", RowBox[List["12", " ", "k"]]]], ")"]], "3"]]]], ")"]]]], SuperscriptBox["64", "k"]]]]]]]]]]]]










Contributed by





G.Huvent (2006)










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





2007-05-02





© 1998- Wolfram Research, Inc.