ArcSin[x] == Limit[Subscript[y, n], n -> Infinity] /; (Subscript[y, n] = Subscript[y, n - 1] + (1/Cos[Subscript[y, n - 1]]) (x - Sin[Subscript[y, n - 1]]) && Subscript[y, 0] = x && Element[x, Reals] && 0 < x < 1)

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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> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <msub> <mi> y </mi> <mi> n </mi> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> y </mi> <mi> n </mi> </msub> <mo> = </mo> <mrow> <mrow> <mrow> <msub> <mi> y </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mfrac> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> y </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> y </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> &#8743; </mo> <msub> <mi> y </mi> <mn> 0 </mn> </msub> </mrow> <mo> = </mo> <mrow> <mi> x </mi> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> x </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsin /> <ci> x </ci> </apply> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <ci> Subscript </ci> <ci> y </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> Set </ci> <apply> <ci> Subscript </ci> <ci> y </ci> <ci> n </ci> </apply> <apply> <ci> Set </ci> <apply> <and /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> y </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sin /> <apply> <ci> Subscript </ci> <ci> y </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> y </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> y </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <and /> <ci> x </ci> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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P. W. Pedersen

2003-08-21