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\usepackage{pandora,euler,amsmath}

\begin{document}

\title{Math 2C, Section A, Spring 1997}
\author{Marcel Oliver}
\date{Midterm Exam I}

\maketitle

\begin{enumerate}

\item Find the limit as $n\to\infty$ for each of the following
sequences: 
\begin{enumerate}
\item $\displaystyle
  a_n = \frac{3 \, \sin (n^2) + 4 \, e^n}{5\,e^{-n} - 6 \, e^n}$
\item $b_n = \left( 1 + \tfrac4n \right)^n$
\item The sequence recursively defined by
\begin{gather*}
  c_0 = 10, \\
  c_{n+1} = \frac{c_n^2 + 3}{2 \, c_n}
\end{gather*}
\end{enumerate} \hfill (10+10+10)

\item Determine whether the given series diverges, converges
conditionally, or converges absolutely.
\begin{enumerate}
\item $\displaystyle \sum_{n=0}^\infty \frac{n!}{2^n}$
\item $\displaystyle \sum_{n=1}^\infty \frac{\sin (5n)}{n^2}$
\item $\displaystyle \sum_{n=2}^\infty \frac{(-1)^n}{n+\ln n}$
\end{enumerate}
\hfill (10+10+10)

\item Show that the series
\[ \sum_{n=2}^\infty \frac{1}{n \, (\ln n)^p} \]
converges when $p>1$ and diverges when $0 \leq p \leq 1$.
\hfill (20)

\item Find the limit of the given series.
\begin{enumerate}
\item $\displaystyle \sum_{n=0}^\infty \left( \frac23 \right)^n $
\item $\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n}$

(Hint: Consider the Taylor series of $\ln(1+x)$.)
\end{enumerate}
\hfill(5+15)

\item Let $\displaystyle f(x) = \frac1{2x-1}$.
\begin{enumerate}

\item Find the Taylor series of $f$ at the point $a=0$ by a method of
your choice.

\item What is the interval of convergence for the power series from
part (a)?
\end{enumerate}
\hfill (10+10)

  
\end{enumerate}

\end{document}
