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\textbf{Math 126: Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Exam II}\quad \textsl{November 9, 2000}
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\bigskip

There are 7 problems on 7 pages worth a total of 84 points.  You start
with 16 points. Each part of a problem is worth the same number of points.

\medskip
You may use a calculator if you wish.

\medskip
To receive partial credit on a problem, you must \emph{show your work
and all important steps}. No credit will be given for an answer if no
work is shown.

\bigskip

\begin{enumerate}

\item (12 pts) Evaluate the integrals.

  \begin{enumerate}[a)]

  \item $\int x e^{2x}\,dx$

  \answerspace

  \item $\int \frac{\sqrt{x^2+1}}{x^4}\,dx$

  \answerspace

  \end{enumerate}

\problemspace
 
\item (6 pts)  Find the partial fraction decomposition of
$\frac{x+2}{x^2(x^2 + 1)}$.

\answerspace

\item (6 pts) Compute the value of the series
$\sum_{n=1}^{\infty} \frac{3^n + 2(-1)^{n+1}}{4^n}$.

\answerspace

\problemspace

\item (12 pts) Find the limit of each sequence, or show it does not exist.

  \begin{enumerate}[a)]

  \item $a_n = \frac{(-1)^n n}{n+1}$

  \answerspace

  \item $a_n = n\,\sin(1/n)$

  \answerspace

  \end{enumerate}

\problemspace

\item (12 pts) Consider the series
$s = \sum_{n=1}^{\infty} \ln\Big(\frac{n}{n+1}\Big)$.

\bigskip

  \begin{enumerate}[a)]

  \item Compute the $4^{th}$ partial sum, $s_4$, exactly.

  \answerspace

  \item Find the value of $s$ or show the series diverges.

  \answerspace

  \end{enumerate}

\problemspace

\item (24 pts) Determine whether the following series converge.

  \begin{enumerate}[a)]

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

  \answerspace

  \item $\sum_{n=2}^\infty \frac {\sqrt{n}}{n^2-1}$

  \answerspace

\newpage

  \item  $\sum_{n=2}^\infty \frac 1{n(\ln(n))^3}$

  \answerspace

  \item $\sum_{n=1}^\infty \frac {5^n}{n^3}$

  \answerspace

  \end{enumerate}

\problemspace

\item (12 pts)

  \begin{enumerate}[a)]

  \item  Show that the series $\sum_{n=1}^\infty (-1)^{n+1}\frac{n!}{(2n-1)!}$
  converges.

  \answerspace

  \item Find an approximate value of the sum of the series with an error
  which is less than $10^{-2}$. Be sure to explain why the error is less
  than $10^{-2}$.

  \answerspace

  \end{enumerate}

\end{enumerate}

\end{document}


