\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\def\R{{\Bbb R}}
\magnification=\magstep1

\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 7\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}


\centerline
{\bf WORKSHEET 7} 

\bigskip 

\item{1.}  {\bf Definition.}  Let $\{a_n\}_{n=1}^\infty$ be a sequence
of real numbers.  We say that the sequence {\it converges} if there is
some real number $L$, called the {\it limit} of the sequence, such
that for any $\varepsilon>0$, there is a number $N$ such that 
$$
|a_n-L|<\varepsilon
$$
whenever $n\geq N$. We sometimes express this by writing
$$
\lim_{n\to\infty}a_n=L.
$$
If there is no such number $L$, then we say that the sequence
$\{a_n\}$ {\it diverges}.

\medskip
\itemitem{a)}  Draw a picture of the real number line which explains
this definition of a convergent sequence.  Take turns explaining it to
each other to make sure that everyone in your group has a clear
understanding. 

\medskip
\itemitem{b)}  Prove that the sequence $\{{1\over n}\}_{n=1}^\infty$
converges.

\medskip
\itemitem{c)}  Give an example of an infinite sequence which does not
converge.  Justify your answer.

\medskip 
\item{2.}  {\bf The Squeeze Theorem.}  Suppose that
$\{a_n\}_{n=1}^\infty$ and $\{c_n\}_{n=1}^\infty$ are sequences which
converge to the same limit $L$, and that $\{b_n\}_{n=1}^\infty$ is a
third sequence such that
$$
a_n\leq b_n\leq c_n \qquad\hbox{ for all $n$.}
$$
Prove that 
$$
\lim_{n\to\infty}b_n=L.
$$

\item{3.}  Prove that the sequence
$$
0.9,\quad0.99,\quad0.999,\quad0.9999,\quad0.99999,\quad0.999999,\quad\ldots
$$
converges.  (to what limit?)

\medskip
\item{4.}  Rewrite the following series using a Riemann sum:
$$
\eqalign{
\pt a & {3\over4}+{5\over9}+{7\over16}+{9\over25}+\cdots=\cr
\pt b & {1\over2}+{2\over3}+{3\over4}+{4\over5}+{5\over6}+\cdots=\cr
\pt c &{101\over3}+{102\over10}+{103\over29}+{104\over66}
+{105\over127}+\cdots=\cr
\pt d & {1\over1}+{1\over3}+{1\over7}+{1\over15}+{1\over31}+\cdots=\cr
\pt e & {2\over3}+{4\over9}+{8\over27}+{16\over81}+{32\over243}+\cdots=\cr}
\hskip2.5in
$$

\medskip
\item{5.}  Let $\{a_n\}_{n=1}^\infty$ be a sequence and define a new sequence $\{S_n\}_{n=1}^\infty$ in the following way:
$$
\eqalign{
S_1&=a_1\cr
S_2&=a_1+a_2\cr
S_3&=a_1+a_2+a_3\cr
S_4&=a_1+a_2+a_3+a_4\cr
   &\>\vdots\cr
S_n&=a_1+a_2+a_3+a_4+\cdots+a_n.\cr}
$$
This is called {\it the sequence of partial sums of $\{a_n\}$}.

\medskip
\itemitem{a)}  Express the sequence in Problem 3 as a sequence of
partial sums.  What are $\{a_n\}$ and $\{S_n\}$ in this case?  What is
$$
\lim_{n\to\infty}a_n
$$
for this sequence of $a_n$?

\medskip
\itemitem{b)}  Now let $\{a_n\}$ be any sequence and suppose that the
corresponding sequence of partial sums $\{S_n\}$ happens to converge.
Write down what that means in terms the definition in Problem 1.

\medskip
\itemitem{c)}  Prove that the sequence $\{a_n\}$ converges to zero.

\medskip
\itemitem{} Hint:  First write down what it is you want to show in
terms of the definition of convergence.  This will keep you on track,
and you can refer back to it if you start to get confused.  Now notice
that (as silly as this may seem) you can write $|a_n|$ as 
$$
|a_n+S_n-S+S-S_n|.
$$
Think of a more helpful way to write $a_n+S_n$ in this expression,
then use the fact (called the {\it triangle inequality}) that
$$
|x+y|\leq|x|+|y|\qquad\forall x,y\in\R.
$$

\medskip
\item{6.}  You are driving along Highway 50 just outside of Canon
City, Colorado eager to get back to Austin for your ESP class the next
day.  Just after you pass the Royal Gorge (Where does that bridge go
to anyway?) and are starting up a steep hill, your fuel pump begins to
fail.  You begin to ascend the hill, but due to the low pressure in
the pump, begin rolling downhill.  You manage to stop descending only
after rolling downhill half the initial distance ascended.  You start
up again and ascend one third of the initial distance upward before
your car acts up again and forces you downhill one fourth of the
initial distance.  This continues so that at the $n^{th}$ stage you
are either rolling downhill one $n^{th}$ of the initial distance or
moving uphill one $n^{th}$ of the initial distance.  If this were to
continue indefinitely, what would happen to your position (that is,
the {\bf net} distance traveled)?  Prove or disprove that the {\it
total} distance traveled is finite.  Do you ever make it up the hill
and back to class?

\medskip
\item{7.}  A bear walks one mile due south, then one mile due east,
and then one mile due north.  He ends up at exactly the same spot
where he started.  What color is the bear?  



\bye