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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Midterm Exam 2 -- October 29, 2004}
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{\it Directions:}  Do all work in the blue exam booklet.  There are
100 possible regular points (distributed as indicated in the questions
below), and 10 Extra Credit points.
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I.  Let $A = \{x \in {\bf R} : 1 < x^2 < 4\}$ and let $B$ be the
interval $[-1,3/2)$.
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\item{A)}  (10) What is the set $S = A\cap B$?
\item{B)}  (10) Find $\sup(S)$ and $\inf(S)$ for the set $S$
from part A.
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II.  
\item{A)} (10) State the Completeness (Least Upper Bound) Axiom for the 
real number system.
\item{B)} (20) Prove that if $x_n$ is a monotone increasing sequence
of real numbers that is bounded above, then $\lim_{n\to\infty} x_n = L$
for some $L \in {\bf R}$.
\item{C)} (5)  Suppose that $x_n$ is a sequence of {\it negative} 
numbers and $x_{n+1}/x_n \le 1$ for all $n \ge 1$.  Show that $x_n$ converges
to some $L \in {\bf R}$.  
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III.
\item{A)} (15) Identify  $\lim_{n\to \infty} x_n$ for the sequence 
$$x_n = {4n\over 7n + 5},$$
and prove that $x_n$ converges to your limit using the $\varepsilon$ - $N$
definition of convergence.
\item{B)} (10)  Use the Limit Theorems for sequences to 
determine
$$\lim_{n\to\infty} {4n^2 + 3n + 1\over 27n^3 + 3}$$
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IV.  Both parts of this question concern the sequence
$$x_n = \cos\left({\pi\sqrt{n}\over 2}\right) + {(-1)^n\over n}$$
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\item{A)} (10) Show that $x_n$ is bounded.
\item{B)} (10) Find two subsequences of $x_n$ converging to two 
different limits.  What does this say about $\lim_{n\to\infty} x_n$?
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{\it Extra Credit.}  (10) Assume that $x_n$ is a sequence that 
converges to $L$ as $n\to \infty$.  Construct a new sequence $y_n$
from the terms of $x_n$ by
$y_n = {x_1 + x_2 + \cdots + x_n\over n}$
Show that $y_n$ also converges to $L$ as $L\to\infty$.  (Hint:
it may help to note that if $x_n \to L$, then there is some
$M$ such that  $|x_n - L| \le M$ for all $n\ge 1$.)
\bye