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\begin{document}


\center{Math 336, Test 1, Spring (1994)}



\title  \endtitle


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 unless otherwise
indicated.  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\noindent{\bf#1.}\ }
\define\prob#1{\par\medskip\noindent\hbox to 3ex{\bf#1.\hfil}}
%\define\fakeprob#1{\par\noindent\phantom{{\bf #1.}}\ }
\define\fakeprob#1{\par\noindent\hbox to 3ex{\hfil}}
\define\ritem#1{\item"(#1)"}
\define\dom{\operatorname{dom}}
\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 25.0pt\relax}

\prob1 (a) Define what it means for a sequence $(f_n)$ of functions on
a set $S$ to converge uniformly to a function $f$ on $S$. (b) Let $S$
be the closed interval $[0,1]$. Give an example of a sequence $(f_n)$
of functions on $S$ such that $f_n$ converges pointwise to a function
$f$ on $S$ but $(f_n)$ does {\bf not} converge uniformly to $f$.
Justify your answer.

\prob2
Let $(f_n)$ be a sequence of continuous functions on a set $S$, and
suppose that $(f_n)$ converges uniformly to a function $f$ 
on $S$. Prove that $f$ is continuous on $S$.

\prob3 
Let $E$ be a subset of a metric space $(S,d)$. Prove that the set $E$
is closed if and only if it contains the limit of every convergent
sequence of points in $E$.


\prob4 Let $\sum a_nx^n$ be a power series with radius of convergence
$R>0$. 
\roster
\ritem a If $0<R_1<R$, show that $\sum a_nx^n$ converges uniformly on
$[-R_1,R_1]$.
\ritem b Use (a) to show that the power series converges to a
continuous function on $(-R,R)$.
\endroster


\prob5  
\roster
\ritem a (5 pts) Calculate the radius of convergence of
$\displaystyle\sum\frac{n^4}{4^n}x^n$.
\ritem b \ourstrut (5 pts) Calculate the radius of convergence of
$\displaystyle\sum\frac{\pi^n}{n!}x^n$.
\ritem c \ourstrut (10 pts) Calculate the {\bf interval of convergence} of
$\displaystyle\sum\frac 1{n^2+1}x^n$. Be sure to handle the endpoints
carefully. Justify your answer.
\endroster

\enddocument



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