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\large\bf Project Based Math 112, Fall 2001\\
How do your Favorite Sequences Stack Up? --- Activity, \S 8.1\\[.25in]
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It's very important to have a feeling for how sequences compare with
each other.  You will examine six sequences, all of which diverge to $\infty$
as $n\!\rightarrow\!\infty$.  You will determine which of these sequences
grow quickly and which grow slowly.

\medskip

We need a way to compare sequences.  Let's write:

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\begin{tabular}{r@{}l}
$\{a_n\}\gg\{b_n\}$ &\ to mean that
$\displaystyle \lim_{n\rightarrow\infty}\frac{a_n}{b_n} = \infty$, \\[.35cm]
and $\{a_n\}\ll\{b_n\}$ &\ to mean that
$\displaystyle \lim_{n\rightarrow\infty}\frac{a_n}{b_n} = 0$. \\
\end{tabular}
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Think of $\gg$ as ``infinitely bigger in the limit'' and $\ll$ as
``infinitely smaller in the limit''.

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{\bf Problem:} Arrange the following sequences, deciding which is ``first'',
``second'', etc., so that:

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first $\ll$ second $\ll$ third $\ll$ fourth $\ll$ fifth $\ll$ sixth
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\begin{tabular}{p{1.5in}p{1.5in}l}
{\bf (a)} $\sqrt{n}$ & {\bf (b)} $e^n$ & {\bf (c)} $n!$ \\[.5cm]
{\bf (d)} $n^2$ & {\bf (e)} $\ln n$ & {\bf (f)} $n^n$ \\
\end{tabular}
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