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\title Math 335, Test 1, 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) State the principle of mathematical induction.   (b) {\sl Using this
principle}, give a careful proof of the formula
$$1+2+\cdots+n=\frac12n(n+1)$$

\prob2 (a) State the Completeness Axiom for the field $\Bbb R$ of real
numbers.  (b) Give an example of a nonempty set $S$ of rational numbers
such that $\operatorname{sup}(S)$ exists and is a real number which is
{\sl not} rational.


\prob3 Let
$s$ and $t$ be real numbers, and let $(s_n)$ and $(t_n)$ be sequences
of real numbers.
\roster
\ritem a Give the definition of what it means for $\lim s_n$ to be equal
to~$s$.
\ritem b Suppose that $\lim s_n=s$ and
$\lim t_n=t$. Prove carefully from the definition that $\lim (s_n+t_n)=s+t$.
\endroster

\prob4  For each of the four sequences $(s_n)_{n=1}^\infty$ below determine whether it
converges and, if it converges, give its limit.  It
is not necessary to give formal proofs that your calculations are
correct.
\roster
\ritem a $s_n=(7n^2-3)/(2n^2+5)$
\ritem b $s_n=(1+1/n)^4$
\ritem c $s_n=\sin(n)/n$
\ritem d $s_n=(2^{n+1}+3)/(2^n+1)$
\endroster

\prob5  Consider the sequence 
$$(s_n)=\left((-1)^n + \frac 1n\right)_{n\ge1}$$
(it might be helpful for you to write out the first few terms to get a
sense of how the sequence behaves).
\roster
\ritem a Compute $\operatorname{sup}\{s_n: n\ge1\}$.
\ritem b Compute $\operatorname{inf}\{s_n: n\ge1\}$.
\ritem c Compute $\lim\,\operatorname{sup}s_n$.
\ritem d Compute $\lim\,\operatorname{inf}s_n$.
\ritem e Does the sequence converge?  Justify your answer.
\endroster



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