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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Exam 2 -- November 4, 2005}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 100 possible regular
points and 10 Extra Credit Points.
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I.  
\item{A)} (20) State and prove the Monotone Convergence Theorem
for sequences.  (You may give the proof in the case that the 
sequence is monotone increasing.)
\item{B)} (10)  Suppose that $x_n$ is a sequence of (strictly) 
{\it negative} numbers and $x_{n+1}/x_n \le 1$ for all $n \ge 1$. 
Show that $\lim_{n\to\infty} x_n = a$ exists in ${\bf R}$
and satisfies $a \le 0$.
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II.   
\item{A)} (10) State the Bolzano-Weierstrass theorem.
\item{B)} (10) Let $x_n = \sin(n)$ (the sequence of values
of $\sin(x)$ at the angles $x$ given by a whole number 
$n\ge 1$ in radian measure).  Show that there exists a subsequence 
$x_{n_k} = \sin(n_k)$ that converges.   
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III. Give an example, or give a short proof that there are no
such examples.
\item{A)} (10)  A convergent series $\sum_{k=1}^\infty a_k$
such that $a_k > 0$ for all $k$ and $\lim_{k\to\infty} ka_k = 4$.
\item{B)} (10)  A convergent series whose sequence of partial sums is
monotone decreasing.  
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IV.  Say whether each of the following infinite series converges absolutely,
converges conditionally, or diverges.  Justify your answer in each case.
\item{A)} (10) $$\sum_{k=1}^\infty {(-1)^{k+1}\over 5k+3}$$
\item{B)} (10) $$\sum_{k=0}^\infty {(-1)^k\over 7^k}$$
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V.  (10)  Using the Ratio Test and other facts as needed, determine
for which $x\in {\bf R}$ the series
$$\sum_{k=0}^\infty {(x-3)^k\over k 6^k}$$
converges.  
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{\it Extra Credit.}  (10) Assume that $x_n$ is a sequence that 
converges to $a$ 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 $a$ as $n\to\infty$.  (Hint:
it may help to recall that if $x_n \to a$, then there is some
$M$ such that  $|x_n - a| \le M$ for all $n\ge 1$.)
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