Pi: Series representations (formula 02.03.06.0009)

Pi

$Mathematica Notation$

Pi

Series representations

Generalized power series

Expansions for Pi

http://functions.wolfram.com/02.03.06.0009.01

Input Form

Pi == Sum[(1/k^3) (-(238/(k + 1)) + 285/(2 (2 k + 1)) - 667/(32 (4 k + 1)) - 5103/(16 (4 k + 3)) + 35625/(32 (4 k + 5))), {k, 1, Infinity}]

Standard Form

Cell[BoxData[RowBox[List["\[Pi]", "\[Equal]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", SuperscriptBox["k", "3"]], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["238", RowBox[List["k", "+", "1"]]]]], "+", FractionBox["285", RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]]]]], "-", FractionBox["667", RowBox[List["32", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "1"]], ")"]]]]], "-", FractionBox["5103", RowBox[List["16", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "3"]], ")"]]]]], "+", FractionBox["35625", RowBox[List["32", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "5"]], ")"]]]]]]], ")"]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mi> π </mi> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> k </mi> <mn> 3 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 285 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 667 </mn> <mrow> <mn> 32 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 5103 </mn> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 35625 </mn> <mrow> <mn> 32 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 238 </mn> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <pi /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 285 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 667 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5103 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 35625 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 238 </cn> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "\[Pi]", "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["-", FractionBox["238", RowBox[List["k", "+", "1"]]]]], "+", FractionBox["285", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]]], "-", FractionBox["667", RowBox[List["32", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "1"]], ")"]]]]], "-", FractionBox["5103", RowBox[List["16", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "3"]], ")"]]]]], "+", FractionBox["35625", RowBox[List["32", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "5"]], ")"]]]]]]], SuperscriptBox["k", "3"]]]]]]]]

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

2001-10-29