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\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