Min[Subscript[x, 1], Subscript[x, 2], \[Ellipsis], Subscript[x, n]] == Limit[((1/n) Sum[Subscript[x, k]^z, {k, 1, n}])^(1/z), z -> -Infinity]

Cell[BoxData[RowBox[List[RowBox[List["Min", "[", RowBox[List[SubscriptBox["x", "1"], ",", SubscriptBox["x", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["x", "n"]]], "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], SubsuperscriptBox["x", "k", "z"]]]]], ")"]], RowBox[List["1", "/", "z"]]], ",", RowBox[List["z", "\[Rule]", RowBox[List["-", "\[Infinity]"]]]]]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> x </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> x </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> x </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msubsup> <mi> x </mi> <mi> k </mi> <mi> z </mi> </msubsup> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <min /> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> n </ci> </apply> </apply> <apply> <limit /> <bvar> <ci> z </ci> </bvar> <condition> <apply> <tendsto /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </apply> </condition> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> k </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Min", "[", RowBox[List[SubscriptBox["x_", "1"], ",", SubscriptBox["x_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["x_", "n_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], SubsuperscriptBox["xx", "k", "z"]]], "n"], ")"]], RowBox[List["1", "/", "z"]]], ",", RowBox[List["z", "\[Rule]", RowBox[List["-", "\[Infinity]"]]]]]], "]"]]]]]]

2001-10-29