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{\large \bf Math 152, Fall, 2004, Workshop 7\par}
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{{\bf Honors Section} 
\par}
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\begin{enumerate}

\item Consider the sequences 
$${\displaystyle
a_{n} =\left(1+\frac{1}{n}\right)^{n}}, \;
{\displaystyle
b_{n} =\left(1+\frac{1}{n^{2}}\right)^{n}}, \;
{\displaystyle
a_{n} =\left(1+\frac{1}{n}\right)^{\sqrt{n}}}.$$

\begin{enumerate}

\item Using your calculator (try the TABLE feature; TblSet is useful, 
too), find and plot the 
first ten terms of each sequence. Using this information, 
try to guess the limit of each 
sequence. 

\item Use L'H\^{o}pital's Rule to find the limit of each sequences. 
(Hint: take logarithms, then 
replace n by $x$ and let $x$ tend to infinity, or 
replace $n$ by $1/x$ and let $x$ tend to zero.) 

\end{enumerate}

\item Suppose that you draw a $2''\times 2''$ square, then join 
the midpoints of its sides to make another square, then 
join the midpoints of that square's sides to form an- 
other square, etc., and that in fact you draw an infinite 
sequence of squares in this way. What is the sum of the 
perimeters of all the squares? 

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\item Two of the following statements are (always) true, and two are
(sometimes) false. Decide which are which. Then for each true statement,
explain briefly why it must be true; for each false statement, give an
example of a series for which it is not true. 

\begin{enumerate}

\item If a series diverges, then its $n$-th term does not approach $0$ as 
$n\to \infty$. 

\item If the $n$-th term of a series does not approach $0$ as $n\to \infty$, 
then the series diverges. 

\item If a series converges, then its partial sums form a bounded sequence. 

\item If the partial sums of a series form a bounded sequence, 
then the series converges. 

\end{enumerate}

\end{enumerate}

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