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\centerline{APPM 1360 \hfill EXAM \#2 \hfill Spr 2003}
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\begin{enumerate}

\vspace{.1in}  
\item
Evaluate the following integrals. (10 points each)
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(a)\lefteqn{\int \sqrt{x} \ln x dx} 
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(b)\lefteqn{\int \frac{dx}{x^2\sqrt{4-x^2}}}
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\item (10 points) Expand  $\frac{1}{(x^3+x)(x^2+x+4)^2}$  in a partial fraction decomposition. 
\textbf{Do not evaluate the coefficients}.
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\item 
Determine whether the following integrals converge or diverge. (10 points each)
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(a) \lefteqn{\int_1^{\infty} \frac{dx}{\sqrt{x^2-0.1}}}
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(b) \lefteqn{\int_{0}^{4} \frac{dx}{(x-3)^2}}
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\item
Consider the sequence $a_n = ( \frac{3^n}{2n+1})^\frac{1}{n}$. 
\begin{enumerate}
\item (8 points) Does the sequence $\{a_n\}$ converge? Explain your answer.
\item (8 points) Does the series $\sum_{n=1}^{\infty}a_n$ converge? Explain your answer.
\end{enumerate}


\item (8 points each) Which of the following converge and which diverge? Justify your answers and if a series converges, find its sum.
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(a) \lefteqn{\sum_{n=1}^{\infty} e^{-n}}
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(b) \lefteqn{\sum_{n=1}^{\infty} \ln{\frac{n}{2n+1}}}
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(c) \lefteqn{\sum_{n=1}^{\infty}\frac{4}{(4n-3)(4n+1)}}



\item (10 points) Solve the initial value problem.
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\lefteqn{(x^2+2)^2 \frac{dy}{dx} + x = x^2 + 2}
\end{enumerate}
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