Pi == Limit[2^(4 n)/(n Binomial[2 n, n]^2), n -> Infinity]

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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> <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> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;n&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <pi /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <ci> Binomial </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29