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\title{Math 1b. Sequences}

\date{Spring 2006}

\begin{document}

\maketitle



\begin{enumerate}



Give a example of a sequence that is: \\
\begin{enumerate}

\item
convergent but not monotone.

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\item
monotone but not convergent.

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\item
bounded but not monotone.

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\item
monotone decreasing and unbounded.

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\item
monotone increasing and convergent.

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\item
unbounded but not monotone.

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\end{enumerate}

\clearpage

\item
Find $a_n$ for the sequence $\displaystyle -\frac{1}{4}, \frac{2}{9}, -\frac{3}{16}, \frac{4}{25}, \ldots$.

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\item
Compute $\lim_{k \to \infty} a_k$ if it exists, where
$$
a_k =\left( 1 + \frac{2}{k} \right)^k
$$


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\item %3
Compute $\lim_{k \to \infty} a_k$ if it exists, where
$$
a_k = \ln k - \ln(3k + 2).
$$

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\item
Show that the sequence defined by
$$
a_1 = 2 \qquad a_{n+1} = \frac{1}{3 - a_n}
$$
satisfies $0 < a_n \leq 2$ and is decreasing.  Deduce that the sequence is convergent and find its limit.

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\end{enumerate}


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