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\begin{document}

\noindent {\bf Math 107 Fall '12 \quad\quad\quad Exam 2 \hfill
\makebox[1.5 in]{Score: \hrulefill}\\

\noindent \makebox[3 in]{Name: \hrulefill} \hfill \makebox[3
in]{TA's Name: \hrulefill}}
\medskip \hrule\smallskip \noindent {\bf
Instructions:} You must show supporting work to receive full and
partial credits. No text book, notes, formula sheets allowed.
\smallskip\hrule
%\begin{center}
%\begin{tabular}[t]{|c|c|c|c|c|c|c|c|}
%\hline No.&\quad 1 \quad & \quad 2 \quad & \quad 3\quad &\quad
%4\quad & \quad 5 \quad &\quad 6\quad &
%\quad Total \quad\\ \hline score & & & & & & & \\
%\hline
%\end{tabular}
%\end{center}

\begin{enumerate}
\item[\bf 1(20pts)] Find the following for the power series: (a) center and radius of
convergence; (b) interval of absolute convergence; (c) points of
conditional convergence if any, (d) interval of convergence; (e)
intervals of divergence,
\[
%\displaystyle{\sum_{k=0}^\infty\frac{(x+2)^k}{2n+1}}
\displaystyle{\sum_{n=0}^\infty\frac{1}{2^n\sqrt{n+1}}(x+2)^n}
\]

\vfill \hfill \makebox[1.5 in]{(\it Continue on Next Page ... \rm
)} \pagebreak

\item[\bf 2(10pts)] A rectangular swimming pool 50 ft long, 20 ft
wide, and 10 ft deep is filled with water to a depth of 9 ft. Set up
an integral to find the work required to pump all of it over the top
rim. (The weight density of water is 62.4 lb per ft$^3$.)



\vskip 4in

\item[\bf 3(10pts)]
%(a) Find the sum of $\displaystyle{\frac{2}{9}+\frac{4}{27}
%+\frac{8}{81}+\cdots +\frac{2\cdot 2^9}{3^{11}}}$.
%\vskip 2in
(a) Find the Taylor polynomial $P_3(x)$ of of $f(x)=\ln{x}$ at point
$x=a=1$.

\vskip 4in

(b) Compare the values of $\ln(1.1)$ and $P_3(1.1)$ by stating the
decimal place to which these two values agree.

\vfill \hfill \makebox[1.5 in]{(\it Continue on Next Page ... \rm )}
\pagebreak
\item[\bf 4(10pts)] (a) Use the Integral Test to determine the value of
$p$ such that $\displaystyle{\sum_{n=2}^\infty\frac{1}{n(\ln n)^p}}$
converges.

\vskip 4.5in

\item[\bf 5(20pts)] Determine if the following series converge.

(a) $\displaystyle{\sum_{n=1}^\infty\cos(\frac{1}{n})}$

\vskip 1.5in

(b) $\displaystyle{\sum_{n=0}^\infty\frac{2^n n!}{(2n)!}}$


\vskip 2in


%Use Taylor's series $\cos(x)=\sum_{n=0}^\infty(-1)^nx^{2n}/(2n)!$ to
%find the limit
%\[ \displaystyle{\lim_{x\to 0}\frac{\cos(\sqrt{x})-1+x/2}{x^2}}
%\]



\vfill \hfill \makebox[1.5 in]{(\it Continue on Next Page ... \rm )}
\pagebreak

\item[\bf 6(20pts)] Use comparison tests to determine if the following series converge
absolutely or diverges:

(a) $\displaystyle{\sum_{n=0}^\infty\frac{2^n\sin n}{3^n+2}}$

\vskip 3in

(b) $\displaystyle{\sum_{n=1}^\infty\frac{n+1}{\sqrt{n(n^2+1)}}}$

\vskip 3in

\item[\bf 7(10pts)] If you use the first 9 terms of the series
$\displaystyle{\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^2}}$ to
approximate it, how many decimal place accuracy do you expect to get
for the approximation? Explain your work. %For what $n$ does the partial sum $S_n$
%approximate the series
%$\displaystyle{\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^2}}$ with a
%margin of error no greater than $10^{-4}$? Show all work.

\end{enumerate}
\vfill \hrule
\medskip \noindent {\bf 2 Bonus Points}: True or False: There are 12 schools in the
Big XII Conference. $\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$
\hfill \makebox[1.0 in]{\hfill (\it ... The End\rm )}
\end{document}
