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Pi




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/02.03.06.0011.01









  


  








Input Form





Pi == (8/(1 + Sqrt[2])) Sum[1/(8 k + 1) - 1/(8 k + 7), {k, 0, Infinity}]








Standard Form





Cell[BoxData[RowBox[List["\[Pi]", "\[Equal]", RowBox[List[FractionBox["8", RowBox[List["1", "+", SqrtBox["2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["8", "k"]], "+", "1"]]], "-", FractionBox["1", RowBox[List[RowBox[List["8", "k"]], "+", "7"]]]]], ")"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mi> &#960; </mi> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 8 </mn> <mtext> </mtext> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <pi /> <apply> <times /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <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> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> k </ci> </apply> <cn type='integer'> 7 </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]", FractionBox[RowBox[List["8", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["8", " ", "k"]], "+", "1"]]], "-", FractionBox["1", RowBox[List[RowBox[List["8", " ", "k"]], "+", "7"]]]]], ")"]]]]]], RowBox[List["1", "+", SqrtBox["2"]]]]]]]]


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





2001-10-29