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Sqrt




Mathematica Notation

Traditional Notation









Elementary Functions > Sqrt[z] > Specific values > Values of nested square roots





http://functions.wolfram.com/01.01.03.0012.01









  


  








Input Form





Sequence[ (1/2) Sqrt[2 - Subscript[b, n - 1] Sqrt[2 + Subscript[b, n - 2] Sqrt[2 + \[Ellipsis] + Subscript[b, 2] Sqrt[2 + Sin[(Pi Subscript[b, 1])/4]]]]] == Sin[Pi (1/2 - Sum[Product[Subscript[b, k - j], {j, 1, k}]/2^k, {k, 1, n}])], Null, ((Subscript[b, k] == -1 || Subscript[b, k] == 1) && 2 <= k <= n && Element[k, Integers]) && Element[Subscript[b, 1], Reals] && -2 <= Subscript[b, 1] <= 2]








Standard Form





Cell[BoxData[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List["2", "-", RowBox[List[SubscriptBox["b", RowBox[List["n", "-", "1"]]], SqrtBox[RowBox[List["2", "+", RowBox[List[SubscriptBox["b", RowBox[List["n", "-", "2"]]], SqrtBox[RowBox[List["2", "+", "\[Ellipsis]", "+", " ", RowBox[List[SubscriptBox["b", "2"], SqrtBox[RowBox[List["2", "+", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", SubscriptBox["b", "1"]]], "4"], "]"]]]]], " "]]]]], " "]]]]], " "]]]]]]], "\[Equal]", RowBox[List["Sin", "[", RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "k"], SubscriptBox["b", RowBox[List["k", "-", "j"]]]]]]]]]]], ")"]]]], "]"]]]], "\[IndentingNewLine]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["b", "k"], "\[Equal]", RowBox[List["-", "1"]]]], "\[Or]", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", "1"]]]], ")"]], "\[And]", RowBox[List["2", "\[LessEqual]", "k", "\[LessEqual]", "n"]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]]]], ")"]], "\[And]", RowBox[List[SubscriptBox["b", "1"], "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["-", "2"]], "\[LessEqual]", SubscriptBox["b", "1"], "\[LessEqual]", "2"]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mfrac> <msub> <mi> b </mi> <mi> n </mi> </msub> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> + </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <mtext> </mtext> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> + </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </msqrt> <mtext> </mtext> </mrow> </mrow> </msqrt> <mtext> </mtext> </mrow> </mrow> </msqrt> <mtext> </mtext> </mrow> </mrow> </msqrt> </mrow> <mo> &#10869; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <msub> <mi> b </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> b </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8744; </mo> <mrow> <msub> <mi> b </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8804; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> &#8804; </mo> <mn> 2 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> &#8230; </ci> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <sin /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <sin /> <apply> <pi /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <or /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <and /> <apply> <leq /> <cn type='integer'> 2 </cn> <ci> k </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <in /> <ci> k </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <reals /> </apply> <apply> <leq /> <cn type='integer'> -2 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List["2", "-", RowBox[List[SubscriptBox["b", RowBox[List["n", "-", "1"]]], " ", SqrtBox[RowBox[List["2", "+", RowBox[List[SubscriptBox["b", RowBox[List["n", "-", "2"]]], " ", SqrtBox[RowBox[List["2", "+", "\[Ellipsis]", "+", RowBox[List[SubscriptBox["b", "2"], " ", SqrtBox[RowBox[List["2", "+", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", SubscriptBox["b", "1"]]], "4"], "]"]]]]]]]]]]]]]]]]]]]]]], "\[Equal]", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "k"], SubscriptBox["b", RowBox[List["k", "-", "j"]]]]]]]]]]], ")"]]]], "]"]]]]]]










References





L. D. Servi, "Nested Square Roots of 2", American Mathematical Monthly, v. 110, issue 4, pp. 326-329 (2003)


Date Added to functions.wolfram.com (modification date)





2007-05-02