\documentclass[12pt]{article}
 
 
\setlength{\parindent}{0pt}
\everymath={\displaystyle}
%\pagestyle{empty}
\newcommand{\answerspace}{\vfill}
\newcommand{\problemspace}{\newpage \hfill\makebox[2.5in][r]{Name:\hrulefill}}
\setlength{\textwidth}{6in}
\setlength{\oddsidemargin}{.30in}
\topmargin=-0.5in
\textheight=9in
 
\begin{document}
 
\textbf{Math 126: Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}
 
\textbf{Exam II}\quad \textsl{November 11, 1999}
\hfill\makebox[2.5in][r]{Section:\hrulefill}
 
\bigskip
 
There are 7 problems worth a total of 80 points
 on 7 pages plus a page to be used as a scratch
sheet.
You start with 20 points.  Unless otherwise noted, each part of a
 problem
is worth the same number of points.
 
\medskip
 
Please read the questions carefully, especially the {\bf bold} words.
 
\medskip
You may use a calculator if you wish.
 
\medskip
If you want partial credit on any problem, be sure to indicate
{\bf clearly} what you are doing.  {\bf Show your work}
and {\bf all} important steps. {\bf No partial credit} will be
given
for any numerical answer that is not explained clearly.
 
\bigskip
 



\begin{enumerate}
 
\item (10 pts)
Give the partial fraction decomposition of
$\frac{x^2-2x+2}{x^3-2x^2+x}$.
 
\answerspace
 
\problemspace


\item (10 pts)
Use a trigonometric substitution to integrate
$\int\frac{x^2}{(x^2+1)^{5/2}}\,dx$.

\answerspace

\problemspace


\item (12 pts) Improper Integrals.
 
  \begin{enumerate}
 
  \item Evaluate $\int_0^{\infty}xe^{-x^2}\,dx$
 
  \answerspace
 
  \item Determine whether the integral
   $\int_1^\infty\frac{\sqrt{x^2+1}}{x^{2}}\,dx$ converges.

  
 \answerspace

  \end{enumerate}
 
\problemspace

\item (12 pts) Infinite Series.

   \begin{enumerate}

   \item Evaluate $\sum _{n=1}^\infty \left( \frac 1{\sqrt n} - \frac
       1{\sqrt{n+1}}\right).$

   \answerspace

   \item Suppose
\begin{math}
  a_1=3,\ a_2=2,
\end{math}
and for any
\begin{math}
  N>2,
\end{math}
\begin{displaymath}
  \sum _{n=1}^N a_n=\frac {N-3\sqrt N}{2N+\sqrt N}.
\end{displaymath}
\end{enumerate}
What is
\begin{math}
  \sum _{n=3}^\infty a_n?
\end{math}
(\textbf{Note:} The range of indices in the question is
\begin{math}
  n=3
\end{math}
to
\begin{math}
  \infty
\end{math}.)
 
\answerspace



\problemspace

\item (18 pts)  Determine whether the following series converge.
 
  \begin{enumerate}

  \item \begin{math}
\sum _{n=1}^\infty \frac 1{\sqrt{n^3+5n-4}}\,.
\end{math}

 \answerspace

  \item $\sum_{n=1}^\infty \frac{n!}{(2n-1)!}$
 
  \answerspace
\problemspace

  \item $\sum_{n=2}^\infty \frac{1}{(\ln(n))^n}$
 
  \answerspace
 
  \end{enumerate}

\item (6 pts)  Determine whether the  series 
$\sum_{n=1}^\infty (-1)^{n+1} \frac {n^2+1}{n^3}$ converges absolutely,
converges conditionally, or diverges. 


\answerspace

\problemspace

\item (12 pts)
 
  \begin{enumerate}
 
  \item Find the interval of convergence of the power series:
  \[ \sum_{n=1}^\infty \frac{(x+1)^n}{2^n}\]
 
  \answerspace
 
  \item Find the Taylor polynomial of order 3 for $f(x) = \ln(2-x)$
   at $x = 1$.
  
  \answerspace

  \end{enumerate}

\end{enumerate}

\problemspace


\end{document}
 
 

