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\title{Math 1b.  Lecture 13 \\
Sequences}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to the notion of a sequence of numbers.

\item
To understand and be able to apply the notions of iteratively defined and recursively defined sequences.

\item
To understand and be able to apply the definitions of convergence and divergence of a sequence.\footnote{We do not cover the formal definition of a convergent sequence.}

\item
To understand a be able to apply the various properties of sequences.

\item
To understand and be able to apply the fact that a bounded monotonic sequence converges.

\end{itemize} 



\section{What is a Sequence?}

A sequence is a list of numbers, $a_1, a_2, a_3, \ldots$.  More formally, we can think of a sequence as a function
$$
a: \mathbb N \rightarrow \mathbb R,
$$
where $a(n) = a_n$.  We often write $\{ a_n \}_{n = 1}^\infty$ for $a_1, a_2, a_3, \ldots$.  We say 
$$
\lim_{n \to \infty} a_n = L
$$
if we can make the terms $a_n$ as close to $L$ as we like by choosing $n$ sufficiently large.

\subsection*{Examples}

\begin{itemize}

\item
$\displaystyle 1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}, \ldots \to 0$

\item
$\displaystyle 1, \frac{1}{2}, \frac{1}{4}, \ldots, \frac{1}{2^{n-1}}, \ldots \to 0$

\item
$\displaystyle 1, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{4}, \ldots, \frac{(-1)^{n+1}}{n}, \ldots \to 0$

\item
$\displaystyle 1, 0, 1, \ldots, \frac{1 + (-1)^{n+1}}{2}, \ldots$ diverges


\item
The Fibonnaci sequence, $1, 1, 2, 3, 5, 8, 13, \ldots$ or 
$f_1 = 1$, $f_2 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n = 3, 4, 5, \ldots$.

\item
$a_n = \sin n$.

\item
$a_0 = 0$ and $a_{n+1} = \sqrt{6 + a_n}$ for $n = 1, 2, 3, \ldots$.


\end{itemize}



\section{Properties of Sequences}

\begin{itemize}

\item
If $\lim_{x \to \infty} f(x) = L$ and $f(n) = a_n$, then
$$
\lim_{n \to \infty} (a_n) = L.
$$

\item
Algebraic properties; e.g., 
$$
\lim_{n\to\infty} (a_n + b_n) = \lim_{n\to\infty} a_n  + \lim_{n\to\infty} b_n.
$$

\item
If $f$ is a continuous function and $\lim_{n \to \infty} a_n = L$, then
$$
\lim_{n \to \infty} f(a_n) = f(L).
$$

\item
The Squeeze Property.

\item
How to apply l'Hospital's Rule.



\end{itemize}


\section{Monotonic and Bounded Sequences}


Every bounded monotonic sequence is convergent.  Consider the sequence $a_0 = 0$ and $a_{n+1} = \sqrt{6 + a_n}$ for $n = 1, 2, 3, \ldots$.
\begin{itemize}

\item
The sequence is bounded by 3 (mathematical induction).  
$$
a_{n+1} = \sqrt{6 + a_n} \leq \sqrt{6 + 3} = 3.
$$

\item
This sequence is monotonic since
\begin{gather*}
0 \geq (a_k - 3) (a_k + 2) = a_k^2 - a_k - 6 \\
\Rightarrow
6 + a_k \geq a_k^2 \\
\Rightarrow
a_{n+1} = \sqrt{6 + a_n} \geq a_k.
\end{gather*}

\end{itemize}
Thus, $\lim_{n \to \infty} a_n = L$.  Since
$$
L = \lim_{n \to \infty} a_{n+1} = \lim_{n \to \infty} \sqrt{6 + a_n} = \sqrt{\lim_{n \to \infty} 6 + a_n} 
=
\sqrt{6 + L}
$$
or $L = 3$.

\section{Worksheet Problems}


\begin{enumerate}


\item
Give a example of a sequence that is:
\begin{enumerate}

\item
convergent but not monotone.




\item
monotone but not convergent.




\item
bounded but not monotone.




\item
monotone decreasing and unbounded.




\item
monotone increasing and convergent.




\item
unbounded but not monotone.



\end{enumerate}



\item
Find $a_n$ for the sequence $\displaystyle -\frac{1}{4}, \frac{2}{9}, -\frac{3}{16}, \frac{4}{25}, \ldots$.



\item
Compute $\lim_{k \to \infty} a_k$ if it exists, where
$$
a_k =\left( 1 + \frac{2}{k} \right)^k
$$






\item %3
Compute $\lim_{k \to \infty} a_k$ if it exists, where
$$
a_k = \ln k - \ln(3k + 2).
$$



\item
Show that the sequence defined by
$$
a_1 = 2 \qquad a_{n+1} = \frac{1}{3 - a_n}
$$
satisfies $0 < a_n \leq 2$ and is decreasing.  Deduce that the sequence is convergent and find its limit.








\end{enumerate}






\section*{References}

\begin{itemize}

\item
\S 8.2 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{March 10, 2006}



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