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\begin{document}





 


\begincenter {Math 336, Test 2, Spring 1996 }\endcenter


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.  Each problem has the point
value 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 (textbooks, 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 15.0pt\relax}
\define\R{\Bbb R}
\define\Q{\Bbb Q} 
\define\Limit{\Cal L}

\prob1 (a) (5 pts) Define what it means for a series to be {\bf
absolutely convergent}.  (b) (15~pts) Prove that if a series is
absolutely convergent, then it converges.

\prob2 (20 pts) Suppose that $(a_n)$ is a sequence of nonnegative
numbers, and that there exists a number $\alpha<1$ such that
$(a_n)^{1/n}\le\alpha$ for all~$n$ sufficiently large. Prove that
$\sum a_n$ converges.

\prob3 (20 pts) Determine the set of nonnegative numbers~$x$  for which the
series
$$\sum \frac{x^n}{2^nn!}$$
converges.


\prob4 (20 pts) Suppose that $(a_n)$ is a sequence of nonnegative numbers and
that $\sum a_n$ converges. Prove that $\sum (a_n)^2$ converges. (Hint:
try comparing one series to the other one.)


\prob5 (20 pts) Give an example of a convergent series $\sum a_n$ such
that $\sum (a_n)^2$ does not converge. (By problem~4, it is impossible
to find an example like this in which all the numbers~$a_n$ are
nonnegative. Try constructing $\sum a_n$ as an alternating series.)

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