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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Information on Exam 2}
\centerline{March 21, 2014}
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\noindent
{\it General Information}
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The second hour exam for the course will be given in class on
Friday, March 28.  This will be an in-class, closed 
book exam.  {\it You may use a calculator, but no other electronic
devices are allowed and no stored text or other saved or online information can
be consulted during the exam.}  I will be happy to hold a late afternoon or evening 
review session to help you prepare.  Late afternoon or evening 
times are possible on Wednesday, March 26 or Thursday, March 27.  
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\noindent
{\it Topics to be Covered}
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The exam will cover the material we have covered since the last 
exam, up through and including class on Friday, March 22.  (This is the content
of Problem Sets 4,5,6.)  (Of course, all 
the material from the first exam about the real number system, least upper 
bounds, completeness, and the technique of proof by mathematical induction 
is relevant here too!)
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\item{1)}  Sequences and convergence.  
\item{2)}  The Limit Theorems for sequences.
\item{3)}  The Monotone Convergence Theorem for sequences and its consequences
(including using it to analyze sequences defined inductively).
\item{4)}  Subsequences and the Bolzano-Weierstrass Theorem.
\item{5)}  The definition of the statement
$\lim_{x\to c} f(x) = L$, variants such as one-sided limits, consequences, limit theorems, 
techniques for computing limits, including use of
sequences to detect when limits do not exist, or to prove
that they do.
\item{6)}  The definition of continuity and its consequences (including
properties of removable, jump and infinite discontinuities).
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\noindent
{\it What to Expect}
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The format will be similar to that of the first exam.
The exam will have four or five questions, each possibly with several
parts.  Some questions will ask for a precise statement of a 
definition or a theorem we have discussed.  Be prepared to give
careful statements of 
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\item{1)}  The definition of convergence for a sequence, and the statement
$\lim_{x\to c} f(x) = L$.  
\item{2)}  The Bolzano-Weierstrass Theorem.
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\noindent
Also know and be able to give these proofs:
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\item{1)}  Part (a) of the Limit Theorem for sequences (Theorem 2.2.5)
(limit of a sum is the sum of the limits).
\item{2)}  The Monotone Convergence Theorem for sequences (Theorem 2.3.3 in 
our text).
\item{3)}  The proof that if $f$ is continuous at $c$ and $\{x_n\}$ is 
a sequence convering to $c$ with $x_n$ in the domain of $f$ for all $n$, 
then $f(x_n)$ converges to  $f(c)$ (Theorem 3.4.13 in text).
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\noindent
{\it Practice Questions}
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\noindent
{\it Don't be concerned about the length of this list.  
The actual exam will be roughly the same length as Exam 1.
The idea is to show the range of different types of questions and
topics that might be covered.}
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\noindent
I.  Give an example of each of the following, or 
give a short proof that there are no such examples:
\item{A)}  A convergent sequence with all strictly negative terms
whose limit is $1$.
\item{B)}  A sequence $\{x_n\}$ such that $\{x_n^2\}$ converges
but $\{x_n\}$ does not.
\item{C)}  Sequences $\{x_n\}$ and $\{y_n\}$ such that $\lim x_n = +\infty$
but $\{x_n y_n\}$ converges to a finite limit.  
\item{D)}  A monotone increasing sequence $\{x_n\}$ that has no
convergent subsequence.  
\item{E)}  A continuous function $f : {\bf R} \to {\bf R}$
such that $f(x) = 0$ if $x \in {\bf Q}$, but $f(\pi) = 3$. 
\item{F)}  A function $f : [a,b] \to {\bf R}$ with $f(a) = 3$, 
$f(b) = -1$, but such 
that $f(x) = 0$ has no solution in $[a,b]$.  
\item{G)}  A function $f$ and a $c$ in the domain of $f$
such that $\lim_{x\to c^+} f(x) = \lim_{x\to c^-}f(x)$, but 
$f$ is not continuous at $c$.  
\item{H)}  A function that is continuous at $x = 0$ 
such that $f(x) \ge 0$ for all $x \ne 0$ but $f(0) = -.00000001$
\item{I)}  A sequence with subsequences converging to 7 different limits (and no others).
\item{J)}  A function that is continuous at $c$ 
for which there is a sequence $x_n \to c$ in the domain of $f$ with $\{f(x_n)\}$ unbounded.  
\item{K)}  A continuous function $f$ on a closed interval $[a,b]$ such that 
$M = {\rm lub}\{f(x) \mid x \in [a,b]\}$ but $M \ne f(x)$ for any $x \in [a,b]$.  
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\noindent
II. Show using the definition that the sequence $x_n = {n + 1\over 3n + 2}$
converges to ${1\over 3}$. 
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\noindent
III.
\item{A)} Show that if $x_n \to a$ and $y_n\to b$, 
then $x_n + y_n \to a+b$.
\item{B)}  Find $\lim_{n\to\infty} \sqrt{n}(\sqrt{9n + 1} - 3\sqrt{n})$.
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\noindent
IV.  Let $\{x_n\}$ be the sequence defined by $x_1  = 1$ and
$x_n = {3x_{n-1} + 1\over 5}$ for all $n \ge 2$.  
\item{A)} Using mathematical induction, show that $x_n \ge 1/2$ for all $n\ge 1$. 
\item{B)} Using mathematical induction, show that $\{x_n\}$ is monotone decreasing. 
\item{C)}  Use part B to show that the sequence
$x_n$ converges to $1/2$.
\item{D)}  An alternate proof that this sequence converges:  Using mathematical induction, show that 
$\left|x_n - {1\over 2}\right| = {1\over 2}\cdot \left({3\over 5}\right)^{n-1}$
for all $n \ge 1$ and use that to show that $x_n \to 1/2$.  
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\noindent
V. Prove the following statements using the $\varepsilon-\delta$
definition of functional limits:
\item{A)}  $\lim_{x\to 3} x^2 - 2x + 4 = 7$.
\item{B)}  $\lim_{x\to e} [x] = 2$  ($[x]$ is the greatest integer
function)
\item{C)}  $\lim_{x\to 2} {1\over x^2} = {1\over 4}$.
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\noindent
VI.  Let $f(x) = x^7 - 5x + 1$.   Prove that $f(x)$ is continuous at all $c\in {\bf R}$.
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\noindent
{\it More Challenging Problems}
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\noindent
You should expect about 10 of the 100 points possible on the exam
to be a problem that is more challenging (to separate the A's from 
the A-'s).  I will also give an 
Extra Credit problem.  This is indicative of the level for those
components on this exam.
VII. In this problem, we say a sequence $\{x_n\}$ is pC (``pseudo-Cauchy'') 
if for every $\varepsilon > 0$, there exists $n_0 \in {\bf N}$ such that
$|x_n - x_{n+1}| < \varepsilon$ for all $n \ge n_0$ (the ``real Cauchy'' 
sequences are the ones for which there exists $n_0$ such that 
$|x_n - x_m| < \varepsilon$ for all $m,n \ge n_0$--be
sure you understand the difference!)
\item{A)}  Show that if $\{x_n\}$ is convergent, then $\{x_n\}$ is pC.
\item{B)}  Consider the sequence defined by $x_n = 1 + {1\over 2} + \cdots + {1\over n}$.
Show that $\{x_n\}$ is pC.
\item{C)}  Consider the subsequence of the sequence in part B
for $n_k = 2^k$.  Then $x_{n_1} = 1 + {1\over 2}$
$x_{n_2} = 1 + {1\over 2} + {1\over 3} + {1\over 4}$, and so forth.
Show that $x_{n_k} > 1 + {k\over 2}$ for all $k \ge 1$.
\item{D)}  From the result in part C, is the sequence in part B convergent?  Explain. 
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\noindent
VIII.  In all parts of this problem, $\{x_n\}$ is a bounded sequence.  
Let $S$ be the set of all real numbers $s$ such that there exists a 
subsequence $\{x_{n_k}\}$ converging to $s$.  ($S$ is the set of {\it cluster points}
of the sequence, as defined in Problem Set 5.)
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\item{A)}  Show that $S$ is bounded above and below.
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\noindent
The number $t = {\rm lub}(S)$ is sometimes called the {\it limit superior}
of the sequence $\{x_n\}$, written
$$t = \lim \sup\ x_n.$$
\item{B)}  What is $\lim \sup\ (-1)^n + {1\over n}$?
\item{C)}  Show that if $\varepsilon > 0$ and $t = \lim \sup\ x_n$, 
then there are only finitely many terms of the sequence $x_n$ with
$x_n > t + \varepsilon$.  
\bye
