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\textbf{Math 126A: Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}
 
\textbf{Exam II}\quad \textsl{November 11, 1999}
 
\bigskip
 
There are 7 problems worth of total of 80 points. You start with 20
points. To receive full credit you must show all your work and include
all important steps.

\medskip You may use a calculator.

\bigskip

\begin{enumerate}
 
\item (10 pts)
Give the partial fraction decomposition of $\frac{x^2-2x+2}{x^3-2x^2+x}$.

\answerspace

\problemspace


\item (10 pts)
Use a trigonometric substitution to integrate
$\int\frac{x^2}{(x^2+1)^{5/2}}\,dx$.

\answerspace

\problemspace

\item (12 pts) Improper Integrals.
 
  \begin{enumerate}
 
  \item Evaluate $\int_0^{\infty}xe^{-x^2}\,dx$
 
  \answerspace
 
  \item Determine whether the integral
   $\int_1^\infty\frac{\sqrt{x^2+1}}{x^{2}}\,dx$ converges.
  
 \answerspace

  \end{enumerate}
 
\problemspace

\item (12 pts) Infinite Series.

   \begin{enumerate}

   \item Evaluate $\sum _{n=1}^\infty \bigg( \frac 1{\sqrt n} - \frac
       1{\sqrt{n+1}}\bigg).$

   \answerspace

   \item Suppose $a_1=3$, $a_2=2$, and for any $N>2$,
   $\sum _{n=1}^N a_n=\frac {N-3\sqrt N}{2N+\sqrt N}$.

   Determine the value of the series $\sum _{n=3}^\infty a_n$. \newline
   (\emph{Note:} The range of indices in the question is $n=3$ to $\infty$.)

\answerspace

\end{enumerate}

\problemspace

\item (18 pts)  Determine whether the following series converge.
 
  \begin{enumerate}

  \item $\sum _{n=1}^\infty \frac 1{\sqrt{n^3+5n-4}}$

 \answerspace

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

\problemspace

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

\item (6 pts)  Determine whether the  series 
$\sum_{n=1}^\infty (-1)^{n+1} \frac {n^2+1}{n^3}$ converges absolutely,
converges conditionally, or diverges. 

\answerspace

\problemspace

\item (12 pts)
 
  \begin{enumerate}
 
  \item Find the interval of convergence of the power series:
  \[ \sum_{n=1}^\infty \frac{(x+1)^n}{2^n}\]
 
  \answerspace
 
  \item Find the Taylor polynomial of order 3 for $f(x) = \ln(2-x)$
   at $x = 1$.
  
  \answerspace

  \end{enumerate}

\end{enumerate}


\end{document}
 
 
