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Posted: Sun Jun 15, 2008 5:21 am
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\begin{document}
\begin{titlepage}
\begin{center}
{\large Nested Radicals} \vspace{1cm} \\
And Other Infinitely Recursive Expressions \vspace{3cm} \\
Michael M$^C$Guffin \vspace{3cm} \\ % Michael McGuffin
{\small {\em prepared} } \\
% \today \vspace{2cm} \\
July 17, 1998 \vspace{2cm} \\
{\small {\em for} } \\
The Pure Math Club \\
University of Waterloo \\
\end{center}
\end{titlepage}
\begin{center}Outline\end{center} \vspace{1.30cm}
1. Introduction \vspace{0.60cm} \\
2. Derivation of Identities \vspace{-0.60cm}
\begin{list}{}{}
\item 2.1 Constant Term Expansions \vspace{-0.60cm}
\item 2.2 Identity Transformations \vspace{-0.60cm}
\item 2.3 Generation of Identities Using Recurrences \vspace{-0.60cm}
\end{list}
3. General Forms \vspace{0.60cm} \\
4. Selected Results from Literature
\pagebreak
{\large 1. Introduction} \\
Examples of Infinitely Recursive Expressions
\linebreak[2]
Series
\[
e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!}
+ \frac{1}{4!} + \ldots
\]
Infinite Products
% Wallis
%
\[ \pi/2 = \frac{2}{1} \times \frac{2}{3} \times \frac{4}{3}
\times \frac{4}{5} \times \frac{6}{5} \times \cdots
\]
Continued Fractions
% Euler
%
\[ e-1 = 1+\frac{2}{2+\frac{3}{3+\frac{4}{4+\frac{5}{5+\ldots}}}} \]
% \[ e = 2 + \frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+
% \ldots
% % \frac{1}{1+\frac{1}{1+\frac{1}{6+\ldots}}}
% }}}}} \]
% Lord Brouncker
%
\[ 4/\pi = 1+\frac{1}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{2+\ldots}}}}
\]
% \[ \pi/2 = 1-\frac{1}{3-\frac{2*3}{1-\frac{1*2}{3-\frac{4*5}{1-
% \frac{3*4}{3-\ldots}
% }}}} \]
Infinitely Nested Radicals (or Continued Roots)
% Kasner number
%
\[ K = \sqrt{1+\sqrt{2+\sqrt{3+\ldots}}} \]
Exponential Ladders (or Towers)
\[ 2 = {(\sqrt{2})}^{{(\sqrt{2})}^{{(\sqrt{2})}^{\cdots}}} \]
Hybrid Forms
% Can be trivially transformed into an exponential ladder.
%
\[ 4 = 2^{\sqrt{2^{\sqrt{2^{\sqrt{2^{\cdots}}}}}}} \]
% This is actually a continued fraction in disguise.
%
\[ \frac{1}{2} = \frac{1}{
\frac{1}{\frac{1}{\ldots}+1+\frac{1}{\ldots}}
+1+
\frac{1}{\frac{1}{\ldots}+1+\frac{1}{\ldots}}
}
\]
\pagebreak
Questions: \vspace{-1.25cm}
\begin{itemize}
\item Does the expression converge ? Are there {\em tests},
or necessary/sufficient conditions for convergence ?
Examples: \vspace{-0.65cm}
\begin{itemize}
\item For series, \vspace{-0.65cm}
\begin{itemize}
\item Terms must go to zero \vspace{-0.65cm}
\item d'Alembert-Cauchy Ratio Test, Cauchy {\em n}th Root Test,
Integral Test, ...
\end{itemize}
\item For infinite products, \vspace{-0.65cm}
\begin{itemize}
\item Terms must go to a value in (-1,1]
\end{itemize}
\item For infinitely nested radicals, \vspace{-0.65cm}
\begin{itemize}
\item Terms can grow ! (But how fast ?)
\end{itemize}
\end{itemize}
\item What does the expression converge to ?
Are there formulae or identities we can use
to evaluate the limit\nolinebreak ?
Example: when $-1 < r < 1$,
\[ \frac{a}{1-r} = a + ar + ar^2 + ar^3 + \ldots \]
\end{itemize}
\pagebreak
{\large 2.1 Constant Term Expansions} \\
Assume that
\[ \sqrt{a+b\sqrt{a+b\sqrt{a+\ldots}}} \]
converges when $a \geq 0$ and $b \geq 0$, and let $L$ be the limit. Then
\[ L = \sqrt{a+b\sqrt{a+b\sqrt{a+\ldots}}} \]
\[ L = \sqrt{a+b L} \]
\[ L^2 - b L - a = 0 \]
\[ L = \frac{b+\sqrt{b^2+4 a}}{2} \]
Hence
\[ \sqrt{a+b\sqrt{a+b\sqrt{a+\ldots}}} = \frac{b+\sqrt{b^2+4 a}}{2} \]
\pagebreak
% paper is 8.5 inches wide
%
\textwidth 8.0in
% margin dimensions are 1 inch less than true value
%
\oddsidemargin -0.75in
\evensidemargin\oddsidemargin
% example code
%
%\topmargin -0.5cm
%\setlength{\topmargin}{0pt}
%\setlength{\topmargin}{-0.8in}
%\setlength{\topskip}{0.3in} % between header and text
%\setlength{\textheight}{9.5in} % height of main text
\begin{document}
\begin{titlepage}
\begin{center}
{\large Nested Radicals} \vspace{1cm} \\
And Other Infinitely Recursive Expressions \vspace{3cm} \\
Michael M$^C$Guffin \vspace{3cm} \\ % Michael McGuffin
{\small {\em prepared} } \\
% \today \vspace{2cm} \\
July 17, 1998 \vspace{2cm} \\
{\small {\em for} } \\
The Pure Math Club \\
University of Waterloo \\
\end{center}
\end{titlepage}
\begin{center}Outline\end{center} \vspace{1.30cm}
1. Introduction \vspace{0.60cm} \\
2. Derivation of Identities \vspace{-0.60cm}
\begin{list}{}{}
\item 2.1 Constant Term Expansions \vspace{-0.60cm}
\item 2.2 Identity Transformations \vspace{-0.60cm}
\item 2.3 Generation of Identities Using Recurrences \vspace{-0.60cm}
\end{list}
3. General Forms \vspace{0.60cm} \\
4. Selected Results from Literature
\pagebreak
{\large 1. Introduction} \\
Examples of Infinitely Recursive Expressions
\linebreak[2]
Series
\[
e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!}
+ \frac{1}{4!} + \ldots
\]
Infinite Products
% Wallis
%
\[ \pi/2 = \frac{2}{1} \times \frac{2}{3} \times \frac{4}{3}
\times \frac{4}{5} \times \frac{6}{5} \times \cdots
\]
Continued Fractions
% Euler
%
\[ e-1 = 1+\frac{2}{2+\frac{3}{3+\frac{4}{4+\frac{5}{5+\ldots}}}} \]
% \[ e = 2 + \frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+
% \ldots
% % \frac{1}{1+\frac{1}{1+\frac{1}{6+\ldots}}}
% }}}}} \]
% Lord Brouncker
%
\[ 4/\pi = 1+\frac{1}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{2+\ldots}}}}
\]
% \[ \pi/2 = 1-\frac{1}{3-\frac{2*3}{1-\frac{1*2}{3-\frac{4*5}{1-
% \frac{3*4}{3-\ldots}
% }}}} \]
Infinitely Nested Radicals (or Continued Roots)
% Kasner number
%
\[ K = \sqrt{1+\sqrt{2+\sqrt{3+\ldots}}} \]
Exponential Ladders (or Towers)
\[ 2 = {(\sqrt{2})}^{{(\sqrt{2})}^{{(\sqrt{2})}^{\cdots}}} \]
Hybrid Forms
% Can be trivially transformed into an exponential ladder.
%
\[ 4 = 2^{\sqrt{2^{\sqrt{2^{\sqrt{2^{\cdots}}}}}}} \]
% This is actually a continued fraction in disguise.
%
\[ \frac{1}{2} = \frac{1}{
\frac{1}{\frac{1}{\ldots}+1+\frac{1}{\ldots}}
+1+
\frac{1}{\frac{1}{\ldots}+1+\frac{1}{\ldots}}
}
\]
\pagebreak
Questions: \vspace{-1.25cm}
\begin{itemize}
\item Does the expression converge ? Are there {\em tests},
or necessary/sufficient conditions for convergence ?
Examples: \vspace{-0.65cm}
\begin{itemize}
\item For series, \vspace{-0.65cm}
\begin{itemize}
\item Terms must go to zero \vspace{-0.65cm}
\item d'Alembert-Cauchy Ratio Test, Cauchy {\em n}th Root Test,
Integral Test, ...
\end{itemize}
\item For infinite products, \vspace{-0.65cm}
\begin{itemize}
\item Terms must go to a value in (-1,1]
\end{itemize}
\item For infinitely nested radicals, \vspace{-0.65cm}
\begin{itemize}
\item Terms can grow ! (But how fast ?)
\end{itemize}
\end{itemize}
\item What does the expression converge to ?
Are there formulae or identities we can use
to evaluate the limit\nolinebreak ?
Example: when $-1 < r < 1$,
\[ \frac{a}{1-r} = a + ar + ar^2 + ar^3 + \ldots \]
\end{itemize}
\pagebreak
{\large 2.1 Constant Term Expansions} \\
Assume that
\[ \sqrt{a+b\sqrt{a+b\sqrt{a+\ldots}}} \]
converges when $a \geq 0$ and $b \geq 0$, and let $L$ be the limit. Then
\[ L = \sqrt{a+b\sqrt{a+b\sqrt{a+\ldots}}} \]
\[ L = \sqrt{a+b L} \]
\[ L^2 - b L - a = 0 \]
\[ L = \frac{b+\sqrt{b^2+4 a}}{2} \]
Hence
\[ \sqrt{a+b\sqrt{a+b\sqrt{a+\ldots}}} = \frac{b+\sqrt{b^2+4 a}}{2} \]
\pagebreak