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\topmatter
\title Math 335, Test 2, Fall (1992) \endtitle
\endtopmatter

The test will be 50 minutes in length.  Please write your name on the
cover of your blue book and write the solutions inside; start the
solution to each problem on a new page.  The problems are worth
20~points apiece; in the case of a problem with multiple sections, the
credit will be divided equally between the parts.  This test
is being administered under the provisions of the Honor Code. Your
work should be your own, and you should not make use of any outside
material (texbooks, notes) during the test.   What you write should be
neat, grammatical, clear and concise. Good luck.
     
\define\prob#1{\par\medskip{\bf#1.}\ }
\define\ritem#1{\item"(#1)"}

\prob1 (a) Define the notion of ``Cauchy sequence''.   (b) Prove that
every convergent sequence is a Cauchy sequence.

\prob2 State and prove the Bolzano-Weierstrass theorem for sequences
of real numbers.

\prob3 Which of the following series converge?
\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 25.0pt\relax}
\roster
\ritem a $\displaystyle \sum\frac{1}{n^4+3n^2}$
\ritem b \ourstrut$\displaystyle \sum\left(\frac 3\pi\right)^n$
\ritem c \ourstrut$\displaystyle \sum\left(\frac \pi3\right)^n$
\ritem d \ourstrut$\displaystyle \sum \frac{(-1)^n\cos(n+1)}{n^{3/2}}$
\endroster

\prob4 For each of the statements below, either give a proof (perhaps
by referring to a theorem) or give a counterexample.
\roster
\ritem a The set $S=[0,1]\cup(2,3]$ is the set of subsequential limits
of some sequence $(s_n)$ of real numbers.
\ritem b A series $\sum a_n$ converges if and only if $\lim a_n=0$.
\endroster

\prob5 Find the set $S$ of subsequential limits of the sequences
$(s_n)$ below:
\roster
\ritem a $(s_n)$ is an enumeration of {\bf all} rational numbers in
$\Bbb R$.
\ritem b 
$$s_n=\cases n, &\text{if $n<10^{10}$}\\
             \ourstrut1+{\displaystyle \frac{(-1)^n}n},&\text{if $n\ge 10^{10}$ and $n$ is
even}\\
             \ourstrut
\sin({\displaystyle\frac{\sqrt 2}n}), &\text{if $n\ge10^{10}$ and $n$ is
odd.}
\endcases
$$
\endroster
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