Pi == 3 Product[Sec[Pi/(12 2^k)], {k, 0, 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> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mi> sec </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> &#960; </mi> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mi> k </mi> </msup> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <pi /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sec /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "\[Pi]", "]"]], "\[RuleDelayed]", RowBox[List["3", " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List["Sec", "[", FractionBox["\[Pi]", RowBox[List["12", " ", SuperscriptBox["2", "k"]]]], "]"]]]]]]]]]]

2001-10-29