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Pi




Mathematica Notation

Traditional Notation









Constants > Pi > Limit representations





http://functions.wolfram.com/02.03.09.0016.01









  


  








Input Form





Pi == Limit[A[n], n -> Infinity] /; A[0] == 4 && B[0] == 1/Sqrt[2] && A[n] == (2 A[n - 1] B[n - 1])/(1 + B[n - 1]) && B[n] == Sqrt[(1 + B[n - 1])/2] && Element[n, Integers] && n > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Pi]", "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List["A", "[", "n", "]"]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["A", "[", "0", "]"]], "\[Equal]", "4"]], "\[And]", RowBox[List[RowBox[List["B", "[", "0", "]"]], "\[Equal]", FractionBox["1", SqrtBox["2"]]]], "\[And]", RowBox[List[RowBox[List["A", "[", "n", "]"]], "\[Equal]", FractionBox[RowBox[List["2", " ", RowBox[List["A", "[", RowBox[List["n", "-", "1"]], "]"]], RowBox[List["B", "[", RowBox[List["n", "-", "1"]], "]"]]]], RowBox[List["1", "+", RowBox[List["B", "[", RowBox[List["n", "-", "1"]], "]"]]]]]]], "\[And]", RowBox[List[RowBox[List["B", "[", "n", "]"]], "\[Equal]", SqrtBox[FractionBox[RowBox[List["1", "+", RowBox[List["B", "[", RowBox[List["n", "-", "1"]], "]"]]]], "2"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> &#960; </mi> <mo> &#63449; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mi> A </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> A </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mn> 4 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> B </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mfrac> <mn> 1 </mn> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> A </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> A </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> B </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> B </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> B </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> B </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[List[], Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <pi /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <ci> A </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> A </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <eq /> <apply> <ci> B </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> A </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> A </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> B </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> B </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> B </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <ci> B </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "\[Pi]", "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["A", "[", "n", "]"]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List["A", "[", "0", "]"]], "\[Equal]", "4"]], "&&", RowBox[List[RowBox[List["B", "[", "0", "]"]], "\[Equal]", FractionBox["1", SqrtBox["2"]]]], "&&", RowBox[List[RowBox[List["A", "[", "n", "]"]], "\[Equal]", FractionBox[RowBox[List["2", " ", RowBox[List["A", "[", RowBox[List["n", "-", "1"]], "]"]], " ", RowBox[List["B", "[", RowBox[List["n", "-", "1"]], "]"]]]], RowBox[List["1", "+", RowBox[List["B", "[", RowBox[List["n", "-", "1"]], "]"]]]]]]], "&&", RowBox[List[RowBox[List["B", "[", "n", "]"]], "\[Equal]", SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["B", "[", RowBox[List["n", "-", "1"]], "]"]]]], ")"]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










Contributed by





Candido Otero Ramos (2007)


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





2007-05-02