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Pi






Mathematica Notation

Traditional Notation









Constants > Pi > Limit representations





http://functions.wolfram.com/02.03.09.0011.01









  


  










Input Form





Pi == Limit[1/Subscript[\[Alpha], n], n -> Infinity] /; Subscript[\[Alpha], n + 1] == (1 + \[Beta]^(n + 1))^4 Subscript[\[Alpha], n] - 2^(2 n + 3) Subscript[\[Beta], n + 1] (1 + Subscript[\[Beta], n + 1] + Subscript[\[Beta], n + 1]^2) && Subscript[\[Beta], n + 1] == (1 - (1 - Subscript[\[Beta], n]^4)^(1/4))/ (1 + (1 - Subscript[\[Beta], n]^4)^(1/4)) && Subscript[\[Alpha], 0] == 6 - 4 Sqrt[2] && Subscript[\[Beta], 0] == Sqrt[2] - 1










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Pi]", "\[Equal]", RowBox[List["Limit", "[", RowBox[List[FractionBox["1", SubscriptBox["\[Alpha]", "n"]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", " ", RowBox[List[RowBox[List[SubscriptBox["\[Alpha]", RowBox[List["n", "+", "1"]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[Beta]", RowBox[List["n", "+", "1"]]]]], ")"]], "4"], " ", SubscriptBox["\[Alpha]", "n"]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "n"]], "+", "3"]]], " ", SubscriptBox["\[Beta]", RowBox[List["n", "+", "1"]]], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["\[Beta]", RowBox[List["n", "+", "1"]]], "+", SubsuperscriptBox["\[Beta]", RowBox[List["n", "+", "1"]], "2"]]], ")"]]]]]]]], "\[And]", RowBox[List[SubscriptBox["\[Beta]", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubsuperscriptBox["\[Beta]", "n", "4"]]], ")"]], RowBox[List["1", "/", "4"]]]]], RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubsuperscriptBox["\[Beta]", "n", "4"]]], ")"]], RowBox[List["1", "/", "4"]]]]]]]], "\[And]", RowBox[List[SubscriptBox["\[Alpha]", "0"], "\[Equal]", RowBox[List["6", "-", RowBox[List["4", " ", SqrtBox["2"]]]]]]], "\[And]", RowBox[List[SubscriptBox["\[Beta]", "0"], "\[Equal]", RowBox[List[SqrtBox["2"], "-", "1"]]]]]]]]]]










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> &#10869; </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> <mfrac> <mn> 1 </mn> <msub> <mi> &#945; </mi> <mi> n </mi> </msub> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> &#945; </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#946; </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msub> <mi> &#945; </mi> <mi> n </mi> </msub> </mrow> <mo> - </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msub> <mi> &#946; </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> &#946; </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </msubsup> <mo> + </mo> <msub> <mi> &#946; </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#946; </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#10869; </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <msubsup> <mi> &#946; </mi> <mi> n </mi> <mn> 4 </mn> </msubsup> </mrow> <mn> 4 </mn> </mroot> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <msubsup> <mi> &#946; </mi> <mi> n </mi> <mn> 4 </mn> </msubsup> </mrow> <mn> 4 </mn> </mroot> </mrow> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#945; </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mrow> <mn> 6 </mn> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#946; </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> </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> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> &#946; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> &#946; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#946; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#946; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> &#946; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#946; </ci> <ci> n </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#946; </ci> <ci> n </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> &#946; </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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[FractionBox["1", SubscriptBox["\[Alpha]", "n"]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[Alpha]", RowBox[List["n", "+", "1"]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[Beta]", RowBox[List["n", "+", "1"]]]]], ")"]], "4"], " ", SubscriptBox["\[Alpha]", "n"]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "n"]], "+", "3"]]], " ", SubscriptBox["\[Beta]", RowBox[List["n", "+", "1"]]], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["\[Beta]", RowBox[List["n", "+", "1"]]], "+", SubsuperscriptBox["\[Beta]", RowBox[List["n", "+", "1"]], "2"]]], ")"]]]]]]]], "&&", RowBox[List[SubscriptBox["\[Beta]", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubsuperscriptBox["\[Beta]", "n", "4"]]], ")"]], RowBox[List["1", "/", "4"]]]]], RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubsuperscriptBox["\[Beta]", "n", "4"]]], ")"]], RowBox[List["1", "/", "4"]]]]]]]], "&&", RowBox[List[SubscriptBox["\[Alpha]", "0"], "\[Equal]", RowBox[List["6", "-", RowBox[List["4", " ", SqrtBox["2"]]]]]]], "&&", RowBox[List[SubscriptBox["\[Beta]", "0"], "\[Equal]", RowBox[List[SqrtBox["2"], "-", "1"]]]]]]]]]]]]










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





2001-10-29





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