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http://functions.wolfram.com/13.05.25.0002.01
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Limit[DivisorSigma[n, m + n]/DivisorSigma[n, n], n -> Infinity] == E^m /;
Element[n, Integers] && n > 0 && Element[m, Integers] && m > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List["DivisorSigma", "[", RowBox[List["n", ",", RowBox[List["m", "+", "n"]]]], "]"]], RowBox[List["DivisorSigma", "[", RowBox[List["n", ",", "n"]], "]"]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "\[Equal]", SuperscriptBox["\[ExponentialE]", "m"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]
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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> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mfrac> <mrow> <msub> <semantics> <mi> σ </mi> <annotation encoding='Mathematica'> TagBox["\[Sigma]", DivisorSigma] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <semantics> <mi> σ </mi> <annotation encoding='Mathematica'> TagBox["\[Sigma]", DivisorSigma] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ⩵ </mo> <msup> <mi> ⅇ </mi> <mi> m </mi> </msup> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <ci> DivisorSigma </ci> <ci> n </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <ci> DivisorSigma </ci> <ci> n </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <ci> m </ci> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List["DivisorSigma", "[", RowBox[List["n_", ",", RowBox[List["m_", "+", "n_"]]]], "]"]], RowBox[List["DivisorSigma", "[", RowBox[List["n_", ",", "n_"]], "]"]]], ",", RowBox[List["n_", "\[Rule]", "\[Infinity]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["\[ExponentialE]", "m"], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
