% Activity on Introducing Infinite Series (adapted from harel's website)
%
\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts,amssymb}
\setlength{\topmargin}{-.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9in}
\pagestyle{plain}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\thispagestyle{empty}

\begin{center}
{\large \begin{bf}
Project Based Math 112, Fall 2001\\
Activity \S 8.1 --- Introduction to Infinite Sequences \\[.25in]
\end{bf}}
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

An infinite sequence (or just ``sequence'') is
an infinite listing of numbers, like

\begin{quotation}
\noindent $ 1, 2, 3, 4, 5, 6, \ldots, $ [the integers]\\
$ 1, 1/2, 1/3, 1/4, \ldots, $ [reciprocals of the integers]\\ 
$ 0, 0, 0, 0, 0, \ldots, $ [all zeros]\\ 
$ 8.17, 1, 2, 3, 3, 3, 3, 3, \ldots $ [some random stuff then all
three's],\\ 
$ 3, 1, 4, 1, 5, 9, 2, \ldots $ [the digits of pi].
\end{quotation}

{\bf 1.} Create your own infinite sequence:\\[.5cm]


{\bf 2.} Create another:\\[.5cm]


{\bf Some notation:} Just as we sometimes like to use constants
like $c$ and $K$ instead of actual numbers like $3$ and $8.64$, there
is a corresponding notation for sequences:

\begin{eqnarray*}
a_1, a_2, a_3, \ldots \ \ \  \text{ or } \ \ \
\{ a_1, a_2, a_3, \ldots \} \ \ \ \text{ or } \ \ \ 
\{ a_n \}_{n=1}^\infty \ \ \  \text{or just} \ \ \ 
\{ a_n \}
\end{eqnarray*}

\vspace{.5cm}
Here $n$ is just an index or ``dummy variable''.
The sequence $\{ a_n\}$ is the same as the sequence $\{a_k\}$.
For example, the sequence $\{ 1/n \}$ is the same as the sequence
$\{1/k\}$, since both of these represent (equal):
\[ 1, 1/2, 1/3, 1/4, \ldots. \]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Part A}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\bf Definition:}
If we can make the terms $a_n$ as close to some number $L$ as we like by
taking $n$ very large, then we say the sequence $\{a_n\}$ tends or
converges to $L$ as $n$ tends to $\infty$, and write
\begin{eqnarray*}
a_n \to  L  \ \  \text{as} \ \  n \to \infty \hspace{1cm} \text{or}
\hspace{1cm}
\lim_{n \to \infty} a_n = L.
\end{eqnarray*}
In this case, we say the sequence is {\em convergent}, or
{\em converges} (to $L$).  Otherwise, we say the sequence {\em
diverges}.\\

For example, the limit of the constant sequence ${c, c, c, c, \ldots}$
is $c$.\\[.1cm]

\newpage 

Given a sequence $\{a_n\}$, there are several possibilities as
to its limit behavior:

\begin{enumerate}
\item[A.] The sequence $\{a_n\}$ may converge to a number $L$.

\item[B.] The sequence may diverge in one of three ways:

\begin{enumerate}
\item[i.] The sequence may  tend to $+\infty$.

\item[ii.] The sequence may  tend to $-\infty$.

\item[iii.] The limit might not exist at all. For example, it might
oscillate, like in $\{cos(n)\}$.
\end{enumerate}
\end{enumerate}
\vspace{.2cm}

\noindent {\bf 3.} Which type is the sequence $1/k$?\\[.2cm]
{\bf 4.} What about $(-1)^n/n$?\\[.2cm]
{\bf 5.} Or $(-1)^k$?\\[1cm]

Now write two (or more) sequences whose convergence is of each of the
types listed above (you may use some of the sequences listed on the first
page).\\[.5cm]

{\bf 6.} Sequences of type A:\\[2cm]

{\bf 7.} Sequences of type Bi:\\[2cm]

{\bf 8.} Sequences of type Bii:\\[2cm]

{\bf 9.} Sequences of type Biii:\\[2cm]


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{Part B}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\bf Another definition}
A sequence  $\{a_n\}$ can also be thought of as a function
$f$ whose domain is the integers $\{1,2,3,\ldots\}$ and whose
range is the real numbers: $f(n) = 1/n$ for example.\\[.2cm]

{\bf 10.} The graph of such a function is an infinite sequence of points.  
Draw one set of axes below, and put in the graphs of the three infinite
sequences (in different colors or dot styles)  given by $f(n) = 1/n, \
g(n)= n, $ and $h(n)= 2-(1/n)$. Be sure to label the units on the
axes.\\[4.5cm]


{\bf 11.} Speaking informally but being as precise as you can, describe
how the graphs above relate to convergence. What visually/geometrically is
key to convergence?\\[2.5cm]


{\bf 12.} Now graph $f(n)=(-1)^n/n$ and $g(n) = (-1)^n$ on the same graph
below:\\[4.5cm]


{\bf 13.} What do these graphs tell you about convergence? Why? Be as
mathematically precise and convincing as you can.\\


\newpage

{\bf Theorem:} \textit{ If $f(x) \to L$ as $x \to \infty$,  and $a_n=f(n)$
for every integer $n$, then $a_n \to L$ as $n \to \infty$.}\\[.5cm]

Thus, since $1/x \to 0$ as $x \to \infty$, then $1/n \to 0$ as $n \to
\infty$. Of course, not every sequence $a_n$ will
be just $f(n)$ for an obvious formula/function $f(x)$.  (For example, let
$a_n$ be the population of the world in the year $n$.) \\[.5cm]

{\bf 14.} For each of the following functions, determine using the theorem
above, whether the sequence defined by the function converges.\\[.5cm]

{\bf a.} $f(n) = n/(n+1)$\\[7cm]


{\bf b.} $g(k) = k \sin(1/k)$\\[7cm]


%%%--------------------------------------------------------------
\newpage
\section{Part C}

{\bf Theorem:}
[Limit laws for convergent sequences] If $a_n \to L$ and $b_n \to M$, then
\begin{itemize}
\item $c a_n \to cL$,
\item $a_n + b_n \to L+M$,
\item $a_n b_n \to L M$,
\item if $M$ is not zero, then $a_n/b_n \to L/M$,
\item $a_n^p \to L^p$ if $p>0$ and $a_n>0$, for all $n$.
\end{itemize}

\vspace{1cm}

{\bf 15.} Use limits you found previously in this activity, and the limit
laws, to find the limits of the following sequences:\\[.5cm]

{\bf a. }$f(n) = 2n/(n+1)$\\[3cm]

{\bf b. }$g(n) = 
\displaystyle{\frac{n}{(n+1)}} +
\displaystyle{\frac{2}{n}}$\\[1cm]



\end{document}
