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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Final Examination}
\centerline{December 15, 2000}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 200 possible 
regular points and 20 possible Extra Credit points (see problem
VII below).
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I.  Let $A = \{\sin(x) : x \in [0,3\pi/4]\}$ and let 
$B = \{x : 1 < x^2 < 4\}$.  
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\item{A)} (10) What is the set $A \cup B$?  
\item{B)} (10) What is the least upper bound of the set 
$C = \{|x - 2| : x \in A\}$?
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II.  
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\item{A)}  (10) State the $\epsilon$ - $n_0$ definition for 
convergence of a sequence.
\item{B)}  (20) Identify $L = \lim_{n\to\infty} x_n$ for
the sequence 
$$x_n = {3n^2 + n\over n^2 + 1}$$
and prove using the definition that $\lim_{n\to\infty} x_n = L$.
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III.  Let $x_n = \sin(n)$ and $y_n = \cos(n)$.  In 
this problem you may use without proof any facts about sequences, 
subsequences, etc. that we proved in class.
\item{A)} (10) Show that
there exists an index sequence of positive integers
$n_k$ such that $x_{n_k} = \sin(n_k)$
converges.   
\item{B)} (10) Does there
exist an index sequence $n_k$ of positive integers such that
{\it both} $x_{n_k}$ and $y_{n_k}$ converge?  Prove your assertion.
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IV.  
\item{A)}  (10) Give the $\epsilon$ - $\delta$ definition for 
the statement $\lim_{x\to c} f(x) = L$.
\item{B)}  (20)  Identify the limit 
$$L = \lim_{x\to 0} x^2 \sin(1/x)$$
and prove using the definition that $\lim_{x\to 0} x\sin(1/x) = L$.
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V.
\item{A)}  (10) State the Intermediate Value Theorem.  
\item{B)}  (10) Let $f(x) = a_1 + a_2\sin(x)$, where $a_1,a_2$ are constants
with $|a_1| < a_2$.  Show that there exists some $c \in [-\pi/2,\pi/2]$
such that $f(c) = 0$.
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VI.  (30)  State and prove both parts of the Fundamental Theorem
of Calculus.
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VII.  True - False.   For each true statement, give a short proof
or reason.  For each false statement give a reason or a counterexample.
Do {\it any five\/} of the following.  If you submit solutions for more 
than five parts, correct answers will be counted as Extra Credit.
In any parts that refer to $f(x)$, use 
$$f(x) = \cases{x^2 & if $x \in [0,1)$\cr
                {1\over 2} & if $x \in [1,2]$\cr}$$
\item{A)} (10)  $f(x)$ is {\it differentiable} at $x = 1$.
\item{B)} (10) Let $\sum_{n=1}^\infty a_n$ be an infinite series with 
positive terms.  If the partial sums $s_N$ are bounded
above, then $\sum_{n=1}^\infty a_n$ converges.
\item{C)} (10)  The {\it conclusion\/} of the Mean Value Theorem
holds for $f(x)$ on the interval $[0,2]$.
\item{D)} (10)  The infinite series
$$\sum_{n=1}^\infty {(-1)^n \over n^3}$$
converges conditionally.
\item{E)} (10)  $f(x)$ is {\it integrable} on $[0,2]$.  (You may use
without proof any general theorems about integrability.)
\item{F)} (10)  The infinite series
$$\sum_{n=1}^\infty {x^n\over n2^n}$$ 
converges for all $x \in {\bf R}$.
\item{G)} (10)  If $p > 1$, then
$${1\over p - 1} \le \sum_{n=1}^\infty {1\over n^p} \le {p\over p-1}$$
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\centerline{\it Have a wonderful holiday season!}
\bye

