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

\textbf{Exam II}\quad \textit{April 20, 2000}

\bigskip

There are 7 pages of questions worth a total of 110 points.
Be sure to show all your work and justify all steps.

\bigskip

\begin{enumerate}

\problemspace

\item (25 pts) Define the following

  \problemspace

  \begin{enumerate}[a)]

  \item $\lim_{x\to \infty} f(x) = L$ and $\lim_{x\to a} f(x) = \infty$.

  \answerspace

  \item $f(x) = o(g(x))$ as $x \to a$.

  \answerspace

  \item Convergence and divergence of a series.

  \answerspace

  \item Absolute and conditional convergence.

  \answerspace

  \end{enumerate}

\newpage

\item (25 pts) State the following theorems precisely.

  \problemspace

  \begin{enumerate}[a)]

  \item L'H\^opital's Rule

  \answerspace

  \item The Limit Comparison Test

  \answerspace

  \item The Integral Test

  \answerspace

  \item The Root Test

  \answerspace

  \item Leibniz's Rule

  \answerspace

  \end{enumerate}

\newpage

\item (15 pts) Compute the following limits, justifying each step.

  \problemspace

  \begin{enumerate}[a)]

  \item $\lim_{x\to0} \frac{x^2\sin(x) - x\sin(x^2)}{\sin(x^5)}$

  \answerspace

  \item $\lim_{n\to\infty} (n+1)^c - n^c$ where $c<1$

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  \item $\lim_{x\to1} x^{1/(1-x)}$

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  \end{enumerate}

\newpage

\item (10 pts) Prove that if $|x|<1$ then $\sum_{n=k}^\infty x^n =
\frac{x^k}{1-x}$.

\newpage

\item (10 pts) Compute the sums of the following series.

  \problemspace

  \begin{enumerate}[a)]

  \item $\sum_{n=1}^\infty \frac{(2^n-1)(2^n+1)}{5^n}$

  \answerspace

  \item $\sum_{n=1}^\infty\frac{1}{4n^2-1}$

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  \end{enumerate}

\newpage

\item (25 pts) Test the following series for convergence or divergence. If
appropriate, determine whether the convergence is absolute or conditional.

  \problemspace

  \begin{enumerate}[a)]

  \item $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^{1+1/n}}$

  \answerspace

  \item $\sum_{n=1}^\infty \frac{n!}{n^n}$

  \answerspace

  \item $\sum_{n=1}^\infty n(-\pi)^{-n}$

  \answerspace

  \newpage

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

  \answerspace

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

  \answerspace

  \end{enumerate}

\end{enumerate}

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