\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 9\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}


\centerline
{\bf WORKSHEET 9} 

\bigskip 

\item{1.}  Test for convergence or divergence.
$$
\eqalign{\pt a&\sum_{n=1}^\infty{n\over n+1}\cr
         \pt e&\sum_{k=1}^\infty e^{-k^2}\cr
         \pt i&\sum_{i=1}^\infty\sin^3\left({1\over i}\right)\cr}\qquad
\eqalign{\pt b&\sum_{n=1}^\infty{\ln n\over 2n^3-1}\cr
         \pt f&\sum_{n=1}^\infty{4n^2-n+3\over n^3+2n}\cr
         \pt j&\sum_{k=1}^\infty{\ln k\over k^2+3}\cr}\qquad
\eqalign{\pt c&\sum_{n=1}^\infty{n\over n^2+1}\cr
         \pt g&\sum_{k=1}^\infty ke^{-k^2}\cr
         \pt k&\sum_{n=1}^\infty{n^2+1\over 3n^2+2n+5}\cr}\qquad
\eqalign{\pt d&\sum_{n=1}^\infty{1\over n\ln n}\cr
         \pt h&\sum_{n=1}^\infty{n+\sqrt n\over 2n^3-1}\cr
         \pt l&\sum_{k=1}^\infty{\log k\over k}\cr}\hskip.9in
$$


\medskip 
\item{2.}  Give an example of each of the following.  If this is not
possible, state why.
$$
\eqalign{
\pt a &\hbox{A function $f$ such that } \int_1^\infty f(x)\,dx \hbox{
diverges, but }\lim_{x\to\infty}f(x)=0.\cr
\pt b &\hbox{A function $f$ such that } \int_1^\infty f(x)\,dx \hbox{
converges, but }\lim_{x\to\infty}f(x)\neq0.\cr
\pt c &\hbox{A series } \sum_{n=1}^\infty a_n \hbox{ which
diverges, but }\lim_{n\to\infty}a_n=0.\cr
\pt d &\hbox{A series } \sum_{n=1}^\infty a_n \hbox{ which
converges, but }\lim_{n\to\infty}a_n\neq0.\cr}\hskip1.8in
$$


\itemitem{3.\hskip12pt a)}  On the same set of axes, sketch the graphs
of the functions $1/x^2$ and $1/(x^2-1)$ over the interval $[2,\infty)$.

\medskip
\itemitem{}  Looking at the graph, do you think that $\int_2^\infty
1/(x^2-1)\,dx$ converges or diverges?  Why?
\medskip
\itemitem{}  Does the {\bf Comparison Test} apply here?

\medskip
\itemitem{b)}  The {\it Ratio Test} for improper integrals
states that if $f(x)$ and $g(x)$ are two positive functions such that 
$$
\lim_{x\to\infty}{f(x)\over g(x)}=L,\qquad0a$ such that whenever
$x>x_0$, 
$$
{L\over2}<{f(x)\over g(x)}<2L.
$$
Why?  Now multiply this entire equation by $g(x)$ and use the
Comparison Test.) 


\eject
\item{4.}  Here is an incorrect statement of the integral test:

\medskip
\item{}  {\bf Theorem.}  Let $f\colon[1,\infty)$ be a function and let
$a_n=f(n)$.  Then
\itemitem{(i)}  if $\int_1^\infty f(x)\,dx$ is convergent, so is the
series $\sum_{n=1}^\infty a_n$, and
\itemitem{(ii)}  if $\int_1^\infty f(x)\,dx$ is divergent, so is the
series $\sum_{n=1}^\infty a_n$.

\medskip
\item{}  What is missing here are some conditions on the function
$f$.  In other words, the integral tes cannot be applied to just any
function.  What are the missing conditions?  Come up with some
functions $f$ for which the above ``Theorem'' is wrong.

\medskip
\item{5.}  Find the sum of the series:
$$
{1\over1\cdot2}+{1\over2\cdot3}+{1\over3\cdot4}+{1\over4\cdot5}+\cdots
$$
Note that the general term $a_n$ is:
$$
a_n={1\over n(n+1)}{\atop.}
$$
What is the partial fractions decompostion of $a_n$?  What does the
series look like if we write the terms that way?



\bye