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{\bf Math 166: Honors Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

{\bf Quiz 10}\quad {\sl Apr. 22, 1999}

\begin{enumerate}
\problemspace
\item 
Determine which of the following statements are true or false.

  \begin{enumerate}[a)]
  \item If $a_n > 0$ and $\sum_{n=1}^\infty a_n$ converges then
  $\lim_{n\to\infty}a_n = 0$.

  \item If $a_n > 0$ and $\lim_{n\to\infty}a_n = 0$ then
  $\sum_{n=1}^\infty a_n$ converges.

  \item If $a_n > 0$ and $\sum_{n=1}^\infty a_n$ converges then
 $\lim_{n\to\infty}a_n^{1/n} = 0$.

  \item If $a_n > 0$ and $\lim_{n\to\infty}a_n^{1/n} = 0$ then
  $\sum_{n=1}^\infty a_n$ converges.

  \item If $a_n>0$ and $\lim_{n\to\infty}a_n=0$, then $\sum_{n=1}^\infty
  (-1)^na_n$ converges.

  \item If $a_n>0$, $b_n>0$ and $\lim_{n\to\infty}\frac{a_n}{b_n} = 0$
  then $\sum_{n=1}^\infty a_n$ converges whenever $\sum_{n=1}^\infty
  b_n$ converges.

  \item If $a_n>0$, $b_n>0$ and $\lim_{n\to\infty}\frac{a_n}{b_n} = 1$
  then $\sum_{n=1}^\infty a_n$ converges if and only if $\sum_{n=1}^\infty
  b_n$ converges.

  \item If $\sum_{n=1}^\infty a_n$ converges, then $\sum_{n=1}^\infty
  |a_n|$ converges.

  \item If $\sum_{n=1}^\infty |a_n|$ converges then $\sum_{n=1}^\infty a_n$
  converges.

  \item If $\sum_{n=1}^\infty a_n$ is conditionally convergent, then
  $\sum_{n=1}^\infty (-1)^na_n$ converges.

  \end{enumerate}

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

  \answerspace

\item  Compute the integral, if possible, or test it for convergence.

  \begin{enumerate}[a)]

  \item $\int_1^\infty \frac{\log(x)}{x^2} dx$ \quad[Hint: $u = \log(x)$]

  \answerspace

 
\item  $\int_0^\infty e^{-x^2}dx$

  \answerspace

  \end{enumerate}

\end{enumerate}

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