\documentclass[12pt]{article}

\hoffset -25truemm
\oddsidemargin=25truemm
\evensidemargin=25truemm
\textwidth=160truemm

\voffset -25truemm
\topmargin=15truemm
\headheight=0truemm
\headsep=0truemm
\textheight=219truemm
\footskip=9truemm

%\addtolength{\topmargin}{-0.2in}
\addtolength{\textheight}{1.0in}

\renewcommand{\baselinestretch}{1.0}

\begin{document}
% \pagestyle{empty}

\newcommand{\ds}{\displaystyle}
\newcommand{\be}{\begin{enumerate}}
\newcommand{\ee}{\end{enumerate}}
\newcommand{\dst}{\displaystyle}

\centerline{\bf\large\bf APPM 1360 --- Fall 1997}
\vskip 8mm
\centerline{\bf\Large\bf Exam 3}
\vskip 8mm
\noindent
ON THE EXAM BOOKLET PLEASE:~~  (1) Write your name,  (2) Student ID,
(3) Lecture Number
(L010 --- Herod, L020 --- Halburd),
and (4) Recitation Instructor.

\noindent
\underline {\bf SHOW ALL WORK}.  A correct answer with no relevant work may
receive no credit, while an incorrect answer accompanied by some correct
work may receive partial credit.
\vskip 3mm
\begin{center}
{\bf Be sure to name any theorem or test you use.}
\end{center}
\begin{center}
\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{center}
\vskip 9mm
\noindent\begin{enumerate}
\item {\em (24 points)}
Which of these series converge?  As always, show all work and name any test
you use.
$$
\mbox{(a)}\ \sum_{n=1}^\infty{{\rm e}^{-n}},\qquad
\mbox{(b)}\ \sum_{n=1}^\infty{(-1)^{n-1}\tan^{-1}n},\qquad
\mbox{(c)}\ \sum_{n=1}^\infty{3^n n!\over (2n)!}.
$$
\item {\em (14 points)}
\begin{enumerate}
\item
Does the following series converge
absolutely,
converge
conditionally, or diverge?
$$
\sum_{n=1}^\infty{(-1)^{n-1}\,{1+n\over n^2}}.
$$
\item
Suppose the sum of the first 99 terms were used as an approximation of the
series in part (a).  Estimate the error in using this approximation.
\end{enumerate}
\item {\em (27 points)}
For what values of $x$ do the following power series
{\it i.} converge absolutely,
{\it ii.} converge conditionally, and {\it iii.} diverge?
State the radius and interval of convergence for each series.
$$
\mbox{(a)}\ \sum_{n=1}^\infty {(-1)^n(4x-1)^n\over 2^{2n}},\qquad
\mbox{(b)}\ \sum_{n=1}^\infty {(x-1)^n\over n^2},\qquad
\mbox{(c)}\ \sum_{n=1}^\infty n!x^n.
$$
\vskip 6cm
\noindent{Continued} $\cdots$
\item {\em (15 points)}
Consider the series 
$$
\sum_{n=1}^\infty a_n
$$
which has $n$th term $a_n$ and $n$th partial sum $S_n$.
For each of the following write TRUE or FALSE.  If a statement is true,
justify your answer by referring to standard
theorems, tests, or definitions.  If it
is false, show that it is false by providing a counter-example and
an explanation of why it is a counter-example.
%\begin{enumerate}
\begin{eqnarray*}
\mbox{(a)} & & \mbox{If }\lim_{n\to\infty} a_n=0
\mbox{ then }\sum_{n=1}^\infty a_n
\mbox{ converges,}\\
\mbox{(b)} & &  \mbox{If }
\lim_{n\to\infty}\sqrt[n]{|a_n|}=0
\mbox{ then }\sum_{n=1}^\infty a_n
\mbox{ converges,}\\
\mbox{(c)} & &  \mbox{If }
\lim_{n\to\infty}S_n=0
\mbox{ then }\sum_{n=1}^\infty a_n
\mbox{ converges.}\\
\end{eqnarray*}
\item {\em (20 points)}
\begin{enumerate}
\item
Find the Taylor polynomial of order 3 generated by $\ln x$ 
expanded about $a=1$.
% \item
% The first few terms in the Maclaurin series for $f(x)$ and $g(x)$ are 
% $$
% f(x)=2+x^2+3x^3+\cdots,\mbox{  and  }
% g(x)=4-x-2x^3+\cdots.
% $$
% Use series multiplication to find the Taylor polynomial of order 3
% generated by $f(x)g(x)$ at $a=0$.
\item Consider the series
$$ \sum_{n=0}^\infty a_n $$
where
$$
a_0=1,\qquad a_{n+1}={\rm e}^x a_n,\quad n\ge 0.
$$
For what values of $x$ does this series converge?  To what function does the
series converge?
\end{enumerate}
\end{enumerate}
\vskip 10mm
\noindent{\bf Extra Credit:}  {\em (10 points)}
Find the first four terms in the Maclaurin series for the solution of
the initial value problem
$$
{d^2y\over dx^2}+xy=0,\qquad
y(0)=1,\quad y'(0)=0.
$$
Hint: Don't even think about trying to solve this equation for $y$ explicitly.

\vskip 1.5cm
\begin{center}
\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{center}
\vskip 1cm
\centerline{\large\bf Some limits}
\vskip 5mm
$$
\lim_{n\to\infty}n^{1/n}=1,\qquad
\lim_{n\to\infty}\left(1+\frac xn\right)^n={\rm e}^x.
$$

\end{document}

















\centerline{\large\bf
A Short Table of Integrals}
\renewcommand{\arraystretch}{1.7}
\[ \begin{array}{lll} \ds
  \int \, du = u + C & & \ds \\ \ds
  \int a \, du = a u + C & & \ds \\ \ds
  \int (du + dv) = \int \, du + \int \, dv & & \ds \\ \ds
  \int u^n \, du = \frac{u^{n+1}}{n+1} + C, ~~ n \neq 1 & & \ds \int
    \frac{du}{u} = \ln |u| + C \\ \ds
  \int \cos u \, du = \sin u + C & & \ds \int \sin u \, du = - \cos
    u + C \\ \ds
  \int \tan u \, du = - \ln |\cos u| + C & & \ds \int \cot u \,
    du = \ln |\sin u| + C \\ \ds
  \int \sec u \, du = \ln | \sec u + \tan u| + C & & \ds \int \csc u \, du
    = - \ln |\csc u + \cot u| + C \\ \ds
  \int e^u \, du = e^u + C & & \ds \int a^u \, du = \frac{1}{\ln a} a^u +
    C, ~~ a > 0, a \neq 1 \\ \ds
  \int \frac{du}{a^2 + u^2} = \frac1a\tan ^{-1} (u/a) + C & & \ds \int
    \frac{du}{\sqrt{a^2-u^2}} = \sin ^{-1} (u/a) + C \\ \ds
  \int \frac{du}{u \sqrt{u^2 - a^2}} = \frac1a\sec ^{-1} |u/a| + C & & \ds \\ \ds
  \int u \, dv = uv - \int v \, du & & \ds
\end{array} \]

$$ $$

\centerline{\bf \large\bf Trigonometric Identities}
$$
\cos^2\theta+\sin^2\theta=1,\qquad
\sin 2\theta = 2\sin\theta\cos\theta,\qquad
\cos2\theta=\cos^2\theta-\sin^2\theta.
$$
\end{document}