Sec[z] == Limit[Sum[(-1)^k/(Pi (1/2 + k) - z), {k, -n, n}], n -> Infinity] /; !Element[z/Pi - 1/2, Integers]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sec", "[", "z", "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "n"]]]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["1", "/", "2"]], "+", "k"]], ")"]]]], "-", "z"]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List[FractionBox["z", "\[Pi]"], "-", FractionBox["1", "2"]]], ",", "Integers"]], "]"]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> sec </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mi> n </mi> </munderover> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> &#8203; </mtext> <mrow> <mrow> <mfrac> <mi> z </mi> <mi> &#960; </mi> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sec /> <ci> z </ci> </apply> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sec", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "n"]]]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "k"]], ")"]]]], "-", "z"]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List["!", RowBox[List[RowBox[List[FractionBox["z", "\[Pi]"], "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]]]]]

2001-10-29