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Pi






Mathematica Notation

Traditional Notation









Constants > Pi > Limit representations





http://functions.wolfram.com/02.03.09.0015.01









  


  










Input Form





Pi == Limit[(2^(n + 1)/(2 - Subscript[b, 1])) ((Subscript[b, n]/2) Sqrt[2 + Subscript[b, n - 1] Sqrt[2 + Subscript[b, n - 2] Sqrt[2 + \[Ellipsis] + Subscript[b, 2] Sqrt[2 + Sin[(Pi Subscript[b, 1])/4]]]]]), n -> Infinity] /; Subscript[b, n] == 1 && Subscript[b, n - 1] == -1 && (Subscript[b, k] == 1 && 2 <= k <= n - 2 && Element[k, Integers]) && Element[Subscript[b, 1], Reals] && -2 <= Subscript[b, 1] <= 2










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Pi]", "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List["n", "+", "1"]]], RowBox[List["2", "-", SubscriptBox["b", "1"]]]], RowBox[List["(", RowBox[List[FractionBox[SubscriptBox["b", "n"], "2"], " ", SqrtBox[RowBox[List["2", "+", RowBox[List[SubscriptBox["b", RowBox[List["n", "-", "1"]]], SqrtBox[RowBox[List["2", "+", RowBox[List[SubscriptBox["b", RowBox[List["n", "-", "2"]]], SqrtBox[RowBox[List["2", "+", "\[Ellipsis]", "+", " ", RowBox[List[SubscriptBox["b", "2"], SqrtBox[RowBox[List["2", "+", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", SubscriptBox["b", "1"]]], "4"], "]"]]]]], " "]]]]], " "]]]]], " "]]]]]]], ")"]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", "\[IndentingNewLine]", RowBox[List[RowBox[List[SubscriptBox["b", "n"], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["b", RowBox[List["n", "-", "1"]]], "\[Equal]", RowBox[List["-", "1"]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["b", "k"], "\[Equal]", "1"]], "\[And]", RowBox[List["2", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["n", "-", "2"]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]]]], ")"]], "\[And]", RowBox[List[SubscriptBox["b", "1"], "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["-", "2"]], "\[LessEqual]", SubscriptBox["b", "1"], "\[LessEqual]", "2"]]]]]]]]










MathML Form







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</mo> <msqrt> <mrow> <mn> 2 </mn> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <mtext> </mtext> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> + </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </msqrt> <mtext> </mtext> </mrow> </mrow> </msqrt> <mtext> </mtext> </mrow> </mrow> </msqrt> <mtext> </mtext> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> b </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 2 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8804; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> &#8804; </mo> <mn> 2 </mn> </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 /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> &#8230; </ci> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <sin /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <leq /> <cn type='integer'> 2 </cn> <ci> k </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <in /> <ci> k </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <reals /> </apply> <apply> <leq /> <cn type='integer'> -2 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </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[RowBox[List[SuperscriptBox["2", RowBox[List["n", "+", "1"]]], " ", RowBox[List["(", RowBox[List[SubscriptBox["b", "n"], " ", SqrtBox[RowBox[List["2", "+", RowBox[List[SubscriptBox["b", RowBox[List["n", "-", "1"]]], " ", SqrtBox[RowBox[List["2", "+", RowBox[List[SubscriptBox["b", RowBox[List["n", "-", "2"]]], " ", SqrtBox[RowBox[List["2", "+", "\[Ellipsis]", "+", RowBox[List[SubscriptBox["b", "2"], " ", SqrtBox[RowBox[List["2", "+", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", SubscriptBox["b", "1"]]], "4"], "]"]]]]]]]]]]]]]]]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", "-", SubscriptBox["b", "1"]]], ")"]], " ", "2"]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List[RowBox[List[SubscriptBox["b", "n"], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["b", RowBox[List["n", "-", "1"]]], "\[Equal]", RowBox[List["-", "1"]]]], "&&", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["b", "k"], "\[Equal]", "1"]], "&&", RowBox[List["2", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["n", "-", "2"]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]]]], ")"]], "&&", RowBox[List[SubscriptBox["b", "1"], "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["-", "2"]], "\[LessEqual]", SubscriptBox["b", "1"], "\[LessEqual]", "2"]]]]]]]]]]










References





L. D. Servi, "Nested Square Roots of 2", American Mathematical Monthly, v. 110, issue 4, pp. 326-329 (2003)










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





2007-05-02





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