GoldenRatio == Limit[Subscript[z, n], n -> Infinity] /; Subscript[z, n + 1] == Sqrt[1 + Subscript[z, n]] && Subscript[z, 0] == 1

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[GoldenRatio]] </annotation> </semantics> <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> <msub> <mi> z </mi> <mi> n </mi> </msub> </mrow> </mrow> <mo> /; </mo> <msub> <mi> z </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> = </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msub> <mi> z </mi> <mi> n </mi> </msub> </mrow> </msqrt> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Set </ci> <apply> <ci> Condition </ci> <apply> <eq /> <ci> GoldenRatio </ci> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> n </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "GoldenRatio", "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[SubscriptBox["z", "n"], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List[RowBox[List[SubscriptBox["z", RowBox[List["n", "+", "1"]]], "\[Equal]", SqrtBox[RowBox[List["1", "+", SubscriptBox["z", "n"]]]]]], "&&", RowBox[List[SubscriptBox["z", "0"], "\[Equal]", "1"]]]]]]]]]]

2001-10-29