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\begin{center}
{\Large Warm-Up Problems and Lecture Problems

March 28, 2003}
\end{center}

\begin{enumerate}

\item  Some properties of sequences are listed below.  Determine which properties, if any, 
	the sequences satisfy.
	\begin{itemize}
	\item  \textbf{Increasing:} $a_{n+1} \geq a_n$ for all $n$.  (i.e., the terms increase.)
	\item  \textbf{Decreasing:} $a_{n+1} \leq a_n$ for all $n$.  (i.e., the terms decrease.)
	\item  \textbf{Monotonic:} The sequence is either increasing or decreasing.
	\item  \textbf{Bounded above:} There is a number $M$ such that $a_n \leq M$ for all $n$.
		(i.e., the sequence is always below a certain number.)
	\item  \textbf{Bounded below:} There is a number $M$ such that $a_n \leq M$ for all $n$.
		(i.e., the sequence is always above a certain number.)
	\item  \textbf{Bounded:} The sequence is bounded above and bounded below.
	\end{itemize}
%
	\begin{enumerate}
	\item  $1,2,3,4,\ldots$
		\\\sol Increasing, monotonic, bounded below.
		\\ $a_n = n$.  $\lim_{n \ra \infty} a_n = \infty$.

	\item  $-1,-2,-3,\ldots$
		\\\sol Decreasing, monotonic, bounded above.
		\\ $a_n = -n$.  $\lim_{n \ra \infty} a_n = -\infty$.

	\item  $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1,5}, \ldots$
		\\\sol Decreasing, monotonic, bounded above, bounded below, bounded.
		\\ $a_n = \frac{1}{n+1}$.  $\lim_{n \ra \infty} a_n = 0$.

	\item  $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots$
		\\\sol Increasing, monotonic, bounded above, bounded below, bounded.
		\\ $a_n = \frac{n}{n+1}$.  $\lim_{n \ra \infty} a_n = 1$.

	\item  $-\frac{1}{2}, \frac{2}{3}, -\frac{3}{4}, \frac{4}{5}, \ldots$
		\\\sol Bounded above, bounded below, bounded.
		\\ $a_n = (-1)^n\frac{n}{n+1}$.  $\lim_{n \ra \infty} a_n$ does not exist.

	\item  $2, 2, 2, 2, 2, \ldots$
		\\\sol Increasing, decreasing, monotonic, bounded above, bounded below, bounded.
		\\ $a_n = 2$.  $\lim_{n \ra \infty} a_n = 2$.

	\item  $1, -2, 3, -4, 5, -6, \ldots$
		\\\sol Satisfies none of the properties.
		\\ $a_n = (-1)^{n+1}n$.  $\lim_{n \ra \infty} a_n$ does not exist.

	\item  $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \ldots$
		\\\sol Decreasing, monotonic, bounded above, bounded below, bounded.
		\\ $a_n = \left(\frac{1}{2}\right)^n$.  $\lim_{n \ra \infty} a_n = 0$.

	\item  $-1, \frac{1}{2}, -\frac{1}{4}, \frac{1}{8}, -\frac{1}{16}, \frac{1}{32}, \ldots$
		\\\sol Bounded above, bounded below, bounded.
		\\ $a_n = \left(-\frac{1}{2}\right)^n = (-1)^n\left(\frac{1}{2}\right)^n$.  
		$\lim_{n \ra \infty} a_n = 0$.

	\end{enumerate}
	If you have time, determine the limit of the above sequences and see
	if you can find a general formula for the $n$-th term of the sequence.


\end{enumerate}


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