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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Information on Exam 2}
\centerline{October 28, 2005}
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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, November 4.  This will be an individual, in-class, closed 
book 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 Wednesday, November 2.
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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.  (Of course, all the material about logic and proofs, including
the technique of proof by mathematical induction, is relevant here too!)
\item{1)}  The Algebraic and Order Limit Theorems for sequences.
\item{2)}  The Monotone Convergence Theorem for sequences and its consequences
\item{3)}  Subsequences and the Bolzano-Weierstrass Theorem.
\item{4)}  Convergence of Infinite series (the definition, key examples
such as geometric series, $\sum_{k=1}^\infty {1\over k^\alpha}$, and 
so forth).
\item{5)}  The Comparison, Alternating Series, and Ratio tests.  
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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 convergence
for an infinite series.
\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 ii of the Algebraic Limit Theorem for sequences (2.3.3)
(limit of a sum is the sum of the limits).
\item{2)}  The Monotone Convergence Theorem for sequences.
\item{3)}  The Comparison Test for infinite series with positive terms.
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\noindent
{\it Practice Questions}
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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 $\sum_{n=1}^\infty x_n y_n$ converges.  
\item{D)}  A monotone increasing sequence $(x_n)$ that has no
convergent subsequence.  
\item{E)}  An absolutely convergent series which is not 
convergent.
\item{F)}  A divergent series whose partial sums are all in 
a bounded interval $[-M,M]$.  
\item{G)}  A sequence that has {\it only one} convergent subsequence.
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\noindent
II.  Find the sum of each of the following infinite series:
\item{A)}  $\sum_{n=3}^\infty \left({-2\over 3}\right)^n$
\item{B)}  $\sum_{n=2}^\infty {1\over n(n-1)}$  (Hint: 
What are $A,B$ to make ${1\over n(n-1)} = {A\over n} + {B\over n-1}$?)
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\noindent
III.  Say whether each of the following series converges absolutely,
converges conditionally, or diverges and justify your answer:
\item{A)}  $$\sum_{k=1}^\infty {(-1)^k\over k^{2/3}}$$
\item{B)}  $$\sum_{k=0}^\infty {(-1)^k 9^k\over (2k)!}$$ (Hint: the Ratio
Test is good for ones like this!)
\item{C)}  $$\sum_{k=0}^\infty {k^2-1\over k^2 + 2k + 4}$$
\item{D)}  $$\sum_{k=1}^\infty {k\over 5^k}$$
\item{E)}  $$\sum_{k=1}^\infty {1\over \sqrt{k} + \sqrt{k+1}}$$
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\noindent
IV.  In this problem, we say a sequence $(x_n)$ is pC (``pseudo-Cauchy'') 
if for every $\varepsilon > 0$, there exists $N \in {\bf N}$ such that
$|x_n - x_{n+1}| < \varepsilon$ for all $n \ge N$ (the ``real Cauchy'' 
sequences are the ones where $|x_n - x_m| < \varepsilon$ for all $m,n \ge N$--be
sure you see the difference!)
\item{A)}  Show that if $(x_n)$ is convergent, then $(x_n)$ is pC.
\item{B)}  Is the converse of the statement in part A true?  Why or why not?
(Hint: Think about sequences of partial sums of infinite series.)
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\noindent
V. ``Limit-comparison'' tests.
\item{A)}  Show that if $a_n > 0$ and $\lim_{n\to\infty} na_n = \ell$ with
$\ell \ne 0$, then $\sum_n a_n$ {\it diverges}.  
\item{B)}  Show that if $a_n > 0$ and $\lim_{n\to\infty} n^2a_n = \ell$ with
$\ell \ne 0$, then $\sum_n a_n$ {\it converges}.
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\noindent
VI.  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 $s$.  
\item{A)}  Show that $S$ is bounded above and below.
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\noindent
The number $t = \sup S$ 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
