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{\bf Math 166: Honors Calculus II}
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{\bf Quiz 9}\quad {\sl Apr. 15, 1999}

\begin{enumerate}
\problemspace
\item Prove the following.
  \begin{enumerate}[a)]

  \item If $a_n > 0$ for all $n$ and if
  $\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = L < 1$, then
  $\sum_{n=1}^\infty a_n$ converges.
  \answerspace
  \answerspace


  \item  If $a_n>0$ for all $n$ and if $\sum_{n=1}^\infty a_n$ converges
  then $\sum_{n=1}^\infty (a_n)^n$ converges.
  \answerspace

  \end{enumerate}


\newpage
\item Determine whether the following series converge.
  \begin{enumerate}[a)]
  \item 
  $\sum_{n=2}^\infty \frac{n-1}{n^{3/2}}$
  \answerspace

  \item 
  $\sum_{n=1}^\infty \left(\frac{\sqrt{n+1}}{n}\right)^n$

  \answerspace

  \problemspace
  \item
  $\sum_{n=1}^\infty \frac{(n!)^3}{n^{4n}}$
  \answerspace

  \end{enumerate}

\end{enumerate}

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