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Pi




Mathematica Notation

Traditional Notation









Constants > Pi > Series representations > Generalized power series > Expansions for 1/Pi





http://functions.wolfram.com/02.03.06.0022.01









  


  








Input Form





1/Pi == 12 Sum[((-1)^k (6 k)! (545140134 k + 13591409))/ (k!^3 (3 k)! (640320^3)^(k + 1/2)), {k, 0, Infinity}]








Standard Form





Cell[BoxData[RowBox[List[FractionBox["1", "\[Pi]"], "\[Equal]", RowBox[List["12", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["6", " ", "k"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["545140134", " ", "k"]], "+", "13591409"]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["k", "!"]], "3"], " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "k"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["640320", "3"], ")"]], RowBox[List["k", "+", FractionBox["1", "2"]]]]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mn> 1 </mn> <mi> &#960; </mi> </mfrac> <mo> &#10869; </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 545140134 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 13591409 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mn> 3 </mn> </msup> <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> <msup> <mn> 640320 </mn> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <factorial /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> k </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 545140134 </cn> <ci> k </ci> </apply> <cn type='integer'> 13591409 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <factorial /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 640320 </cn> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox["1", "\[Pi]"], "]"]], "\[RuleDelayed]", RowBox[List["12", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["6", " ", "k"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["545140134", " ", "k"]], "+", "13591409"]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "3"], " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "k"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["640320", "3"], ")"]], RowBox[List["k", "+", FractionBox["1", "2"]]]]]]]]]]]]]]]


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





2001-10-29