\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 10\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}


\centerline
{\bf WORKSHEET 10} 

\bigskip 

\item{1.}   Carefully state the following tests for convergence or
divergence of a series.  Give examples of series to which the tests
could be applied in determining convergence or divergence.
$$
\eqalign{\pt a& Divergence\>Test\cr
         \pt d& Integral\>Test\cr}\qquad
\eqalign{\pt b& Geometric\>Series\>Test\cr
         \pt e& Alternating\>Series\>Test\cr}\qquad
\eqalign{\pt c& Comparison\>Test\cr
         \pt f& p-Series\>Test\cr}\hskip.3in
$$


\itemitem{2.\hskip12pt a)}  State the {\it comparison test} for the 
convergence or divergence of a series.  Apply this test to the
following series:
$$
\pt i     \sum_{n=1}^\infty {4n   \over n^3+n}\qquad
\pt {ii}  \sum_{n=1}^\infty {4n   \over n^3-n}\qquad
\pt {iii} \sum_{n=1}^\infty {4n^2 \over n^3+n}\hskip1.7in
$$

\itemitem{b)}  In which of these examples was the application of the
comparison test more difficult.  Why?

\medskip
\itemitem{c)}  For each of these series, give a quick intuitive
argument which may not be completely rigorous but which sidesteps the
difficulties you encountered.  (Hint: What is the {\bf general}
behavior of the terms as $n\to\infty$?)  Why is this argument not
rigorous? 

\medskip
\itemitem{d)}  Determine $\lim_{n\to\infty}n^2a_n$ where $a_n$ is the
$n^{\rm th}$ term of series ii) in part a) above. 

\medskip
\itemitem{e)}  Use your answer to c) to show that there must be an
integer $N$ such that
$$
a_n<{1\over n^2}
$$
for all $n\geq N$.  What does this tell you about the convergence of
the series $\sum a_n$? 

\medskip
\itemitem{f)}  Now we will formalize this argument into a process that
will allow us to apply the comparison test more easily in many cases.
Suppose that $\sum_{n=1}^\infty p_n$ is a series of positive terms to
be tested for convergence or divergence.  Suppose, also, that we have
another series $\sum_{n=1}^\infty c_n$ which we know converges and
that the limit
$$
\lim_{n\to\infty}{p_n\over c_n}=a
$$
exists.  Show that there must be an integer $N$ such that
$$
p_n<(a+1)c_n
$$
for all $n\geq N$.  What does this tell you about the convergence of
the series $\sum p_n$?

\eject
\item{}  In part f) you proved part of the {\it limit-comparison
test}.  Fully stated, it says:  

\medskip
\item{}  Let $\sum_{n=1}^\infty p_n$ be a series of positive terms
to be tested for convergence or divergence. 

\smallskip
\itemitem{i)}  Let $\sum_{n=1}^\infty c_n$ be a convergent series of
positive terms.  If $\lim_{n\to\infty}p_n/c_n$ exists, then
$\sum_{n=1}^\infty p_n$ also converges.

\smallskip
\itemitem{ii)}  Let $\sum_{n=1}^\infty d_n$ be a divergent series of
positive terms.  If $\lim_{n\to\infty}p_n/d_n$ exists and is not zero
or if it is infinite, then $\sum_{n=1}^\infty p_n$ also diverges.

\medskip
\itemitem{g)}  Apply the limit-comparison test to each of the series
in part a).

\medskip
\item{3.}  The first thing you should do (at least mentally if not on
paper) when trying to determine whether a series $\sum_{n=1}^\infty
a_n$ converges or diverges is to consider the limit 
$$
\lim_{n\to\infty}a_n.
$$
The primary information you gain here is that if the limit is not
zero, then the series diverges, and if the limit is zero, then it is
{\bf possible} that the series converges.  (Why do you think I
stressed the word ``possible''?)

\medskip
\item{}  In the case that the limit is zero (so that we must explore
further tests), there is also some secondary information that can be
gained from considering this limit.  Specifically, noticing the
general behavior of the terms as $n\to\infty$ can often suggest what
series to use for a comparison or limit-comparison test.  For example,
the terms of the series in Problem 2 a) tend to behave as $4/n^2$,
$4/n^2$, and $4/n$ respectively, as $n$ tends to infinity.

\medskip
\itemitem{a)}  For each of the following series, find the limit of the
sequence of terms.
$$
\eqalign{
\pt {i} & \sum_{n=1}^\infty { 4n^2-7n+\pi \over {1\over97}n^3-80n^2-10^5n-1}\cr
\pt {iv} & \sum_{n=1}^\infty n^{-{n+1 \over n}}\cr}\qquad
\eqalign{
\pt {ii} & \sum_{n=1}^\infty { \sin{1/n}\over\ln n }\cr
\pt {v} & \sum_{n=1}^\infty n^{-{n \over n+1}}\cr}\qquad
\eqalign{
\pt {iii} & \sum_{n=1}^\infty 6n^{38}e^{.005n}\cr
\pt {vi} & \sum_{n=1}^\infty {\ln n \over n^{1.001}}\cr}\hskip.1in
$$

\medskip
\itemitem{b)}  For each of the series given in part a), describe the
limiting behavior of the terms.

\medskip
\itemitem{c)}  Use your answer to part b) to apply either the
comparison test or limit-comparison test to each of these series.

\medskip
\item{4.}  Suppose $a_n\geq0$ and ${^\infty\atop{\sum\atop_{n=1}}}a_n^2$
converges.  Show that ${^\infty\atop{\sum\atop_{n=1}}}{a_n\over n}$
converges.
\item{}
\item{} (Hint: compare $a_n\over n$, $a_n^2$, $1\over n^2$)






\bye