Pi: Series representations (formula 02.03.06.0020)

Pi

$Mathematica Notation$

Pi

Series representations

Generalized power series

Expansions for 1/Pi

http://functions.wolfram.com/02.03.06.0020.01

Input Form

1/Pi == ((2 Sqrt[2])/9801) Sum[((4 k)! (26390 k + 1103))/(k!^4 396^(4 k)), {k, 0, Infinity}]

Standard Form

Cell[BoxData[RowBox[List[FractionBox["1", "\[Pi]"], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", SqrtBox["2"]]], " "]], "9801"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "k"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["26390", " ", "k"]], "+", "1103"]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["k", "!"]], "4"], " ", SuperscriptBox["396", RowBox[List["4", " ", "k"]]]]]]]]]]]]]]

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> π </mi> </mfrac> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mn> 9801 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 26390 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1103 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msup> <mn> 396 </mn> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> </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 /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 9801 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 26390 </cn> <ci> k </ci> </apply> <cn type='integer'> 1103 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 396 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </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]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox["2"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "k"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["26390", " ", "k"]], "+", "1103"]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "4"], " ", SuperscriptBox["396", RowBox[List["4", " ", "k"]]]]]]]]]], "9801"]]]]]

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

2001-10-29