ArithmeticGeometricMean[1, b] == Product[(1/2) (1 + Subscript[q, k]), {k, 0, Infinity}] /; Subscript[q, 0] == b && Subscript[q, k + 1] == (2 Sqrt[Subscript[q, k]])/(1 + Subscript[q, k])

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["1", ",", "b"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List["1", "+", SubscriptBox["q", "k"]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["q", "0"], "\[Equal]", "b"]], "\[And]", RowBox[List[SubscriptBox["q", RowBox[List["k", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["2", SqrtBox[SubscriptBox["q", "k"]]]], RowBox[List["1", "+", SubscriptBox["q", "k"]]]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mi> agm </mi> <annotation-xml encoding='MathML-Content'> <ci> ArithmeticGeometricMean </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> q </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> q </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mi> b </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> q </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#10869; </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <msub> <mi> q </mi> <mi> k </mi> </msub> </msqrt> </mrow> <mrow> <msub> <mi> q </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> ArithmeticGeometricMean </ci> <cn type='integer'> 1 </cn> <ci> b </ci> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> q </ci> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> q </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> q </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> q </ci> <ci> k </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> q </ci> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["1", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["q", "k"]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["q", "0"], "\[Equal]", "b"]], "&&", RowBox[List[SubscriptBox["q", RowBox[List["k", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["2", " ", SqrtBox[SubscriptBox["q", "k"]]]], RowBox[List["1", "+", SubscriptBox["q", "k"]]]]]]]]]]]]]]

2007-05-02