ArithmeticGeometricMean[1, b] \[Proportional] SeriesData[$CellContext`z, 0, {1, 0, Rational[1, 6] $CellContext`m, 0, Rational[1, 120] (-4 $CellContext`m + 9 $CellContext`m^2), 0, Rational[1, 5040] (16 $CellContext`m - 180 $CellContext`m^2 + 225 $CellContext`m^3), 0, Rational[1, 362880] (-64 $CellContext`m + 3024 $CellContext`m^2 - 12600 $CellContext`m^3 + 11025 $CellContext`m^4)}, 1, 11, 1] /; (b -> 1)

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["1", ",", "b"]], "]"]], "\[Proportional]", InterpretationBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", "-", "1"]], "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "2"], "16"], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "3"], "32"], "-", FractionBox[RowBox[List["21", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "4"]]], "1024"], "+", FractionBox[RowBox[List["31", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "5"]]], "2048"], "-", FractionBox[RowBox[List["195", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "6"]]], "16384"], "+", FractionBox[RowBox[List["319", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "7"]]], "32768"], "-", FractionBox[RowBox[List["34325", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "8"]]], "4194304"], "+", FractionBox[RowBox[List["58899", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "9"]]], "8388608"], "-", FractionBox[RowBox[List["410771", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "10"]]], "67108864"], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "11"], "]"]]]], SeriesData[$CellContext`z, 0, List[1, 0, Times[Rational[1, 6], $CellContext`m], 0, Times[Rational[1, 120], Plus[Times[-4, $CellContext`m], Times[9, Power[$CellContext`m, 2]]]], 0, Times[Rational[1, 5040], Plus[Times[16, $CellContext`m], Times[-1, 180, Power[$CellContext`m, 2]], Times[225, Power[$CellContext`m, 3]]]], 0, Times[Rational[1, 362880], Plus[Times[-64, $CellContext`m], Times[3024, Power[$CellContext`m, 2]], Times[-1, 12600, Power[$CellContext`m, 3]], Times[11025, Power[$CellContext`m, 4]]]]], 1, 11, 1]]]], "/;", RowBox[List["(", RowBox[List["b", "\[Rule]", "1"]], ")"]]]]]]

<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> &#8733; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mn> 16 </mn> </mfrac> <mo> + </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mn> 32 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 21 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mn> 1024 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 31 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 5 </mn> </msup> </mrow> <mn> 2048 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 195 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> </mrow> <mn> 16384 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 319 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 7 </mn> </msup> </mrow> <mn> 32768 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 34325 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 8 </mn> </msup> </mrow> <mn> 4194304 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 58899 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 9 </mn> </msup> </mrow> <mn> 8388608 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 410771 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 10 </mn> </msup> </mrow> <mn> 67108864 </mn> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 11 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> ArithmeticGeometricMean </ci> <cn type='integer'> 1 </cn> <ci> b </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 16 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 32 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 21 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 1024 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 31 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <cn type='integer'> 2048 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 195 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <cn type='integer'> 16384 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 319 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <cn type='integer'> 32768 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 34325 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <cn type='integer'> 4194304 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 58899 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 9 </cn> </apply> <apply> <power /> <cn type='integer'> 8388608 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 410771 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 10 </cn> </apply> <apply> <power /> <cn type='integer'> 67108864 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 11 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["1", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[InterpretationBox[RowBox[List["$CellContext`z", "+", RowBox[List[FractionBox["1", "6"], " ", "$CellContext`m", " ", SuperscriptBox["$CellContext`z", "3"]]], "+", RowBox[List[FractionBox["1", "120"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "$CellContext`m"]], "+", RowBox[List["9", " ", SuperscriptBox["$CellContext`m", "2"]]]]], ")"]], " ", SuperscriptBox["$CellContext`z", "5"]]], "+", RowBox[List[FractionBox["1", "5040"], " ", RowBox[List["(", RowBox[List[RowBox[List["16", " ", "$CellContext`m"]], "-", RowBox[List["180", " ", SuperscriptBox["$CellContext`m", "2"]]], "+", RowBox[List["225", " ", SuperscriptBox["$CellContext`m", "3"]]]]], ")"]], " ", SuperscriptBox["$CellContext`z", "7"]]], "+", RowBox[List[FractionBox["1", "362880"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "64"]], " ", "$CellContext`m"]], "+", RowBox[List["3024", " ", SuperscriptBox["$CellContext`m", "2"]]], "-", RowBox[List["12600", " ", SuperscriptBox["$CellContext`m", "3"]]], "+", RowBox[List["11025", " ", SuperscriptBox["$CellContext`m", "4"]]]]], ")"]], " ", SuperscriptBox["$CellContext`z", "9"]]], "+", InterpretationBox[SuperscriptBox[RowBox[List["O", "[", "$CellContext`z", "]"]], "11"], SeriesData[$CellContext`z, 0, List[], 1, 11, 1], Rule[Editable, False]]]], SeriesData[$CellContext`z, 0, List[1, 0, Times[Rational[1, 6], $CellContext`m], 0, Times[Rational[1, 120], Plus[Times[-4, $CellContext`m], Times[9, Power[$CellContext`m, 2]]]], 0, Times[Rational[1, 5040], Plus[Times[16, $CellContext`m], Times[-1, 180, Power[$CellContext`m, 2]], Times[225, Power[$CellContext`m, 3]]]], 0, Times[Rational[1, 362880], Plus[Times[-64, $CellContext`m], Times[3024, Power[$CellContext`m, 2]], Times[-1, 12600, Power[$CellContext`m, 3]], Times[11025, Power[$CellContext`m, 4]]]]], 1, 11, 1], Rule[Editable, False]], "/;", RowBox[List["(", RowBox[List["b", "\[Rule]", "1"]], ")"]]]]]]]]

2001-10-29