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\textbf{Math 126: Calculus II}
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\textbf{Exam II}\quad \textsl{November 14, 2002}
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\bigskip

There are 6 problems on 7 pages (including the cover page)
 worth a total of  90 points.  You start
with 10 points. Unless otherwise indicated,
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}
including
\emph{all important steps}. No credit will be given for an answer if no
work is shown.

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\

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\begin{enumerate}





\item (15 points)
Let $C$ be the curve $y=\cos{x}$ for $0 \leq x \leq  \frac{\pi}{2}$.
    \begin{enumerate}
    \item Set up, but do not evaluate, a definite integral for the
    arc length of $C$.
    \answerspace

    \item Set up, but do not evaluate, a definite integral for the
    area of the surface of revolution obtained by rotating $C$ about
    the $x$-axis.
    \answerspace
    \end{enumerate}





\problemspace





\item  (10 points)
Solve the initial value problem
\[ x^{2} y^{'} + 4xy +1 = 0, \hspace{20pt} y(1)=0. \]

\answerspace

\problemspace

 \item (20 points) 
   \begin{enumerate}
  \item  Determine whether the \textbf{sequence}
  \[ a_{n}=\frac{\ln (n+3)}{(n+3)} \]
  converges or diverges. If it   converges, find the limit.

 \answerspace  

\ 
\answerspace


   \item Determine whether  the series 
     \begin{displaymath}
       \sum_{n=1}^{\infty} \frac{\ln (n+3)}{(n+3)^2}
     \end{displaymath}
   is convergent or divergent.


 \answerspace
\ 

\answerspace

\ 

\answerspace

  \end{enumerate}

\problemspace


\item  (20 points)
  \begin{enumerate}
  \item 
  
Find $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{8^{n-1}}{3^{2n}}$.
\answerspace

\item
Determine whether the series

 $\sum_{n=1}^{\infty} (-1)^n  \frac{n^2}{4^n}$
is convergent or divergent.

\answerspace

\end{enumerate}

\problemspace

\item (15 points)
Determine whether the series
\[ \sum_{n=2}^{\infty} (-1)^n \frac{\sqrt{n+1}}{n} \]
is absolutely convergent, 
conditionally convergent or divergent.

\answerspace

\problemspace


\item (10 points)
   Determine whether
  \[ \int_{1}^{\infty} \frac{2+\cos{x}}{x} \, dx \]
  is convergent or divergent. 
\answerspace

\answerspace

\end{enumerate}

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