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\center Math 336, Final Exam, Spring (1994) }

The test will be 2 hours 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.
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.}}\ }
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\define\ritem#1{\item"(#1)"}
\define\dom{\operatorname{dom}}
\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 15.0pt\relax}

\prob1 (30 pts) Write an essay at least two pages in length on the
theory of integration. State the major definitions in the theory,
describe the major theorems, and give the proof of at least one of
these theorems.

\prob2 (a) (10 pts) Define what it means for a subset of ${\Bbb R^n}$
to be compact. (b) (10~pts) Give an example of a subset of $\Bbb R^2$
which is {\bf not} compact. Justify your answer.

\prob3 Let $(f_n)$ be a sequence of functions on the
interval $[a,b]$. 
\roster \ritem a (5 pts)  Define what it means for the sequence
$(f_n)$ to converge {\bf pointwise} on $[a,b]$.
\ritem b (5 pts) Define what it means for the sequence $(f_n)$ to
converge {\bf uniformly} on $[a,b]$.
\ritem c (10 pts) Give an example of a sequence $(f_n)$ which
converges pointwise on $[a,b]$ but not uniformly. Justify your answer.
\endroster

\prob4 (20 pts) Let $f(x)=\displaystyle\sum^\infty_{n=1}\frac{x^n}{n^2}$.
Determine the largest interval on which $f$ represents a continuous
function.


\prob5 This problem has four basic parts, worth 10 points each.  There
is also an extra-credit portion, worth 10 points.  

\smallskip
\noindent (a)  State Taylor's theorem.

\smallskip
\noindent (b) Let $n\ge1$ be an integer. 
Use Taylor's theorem to show that 
there is a number $c_n$ between $n$ and $n+1$ such that
$$\ln(n+1)=\ln(n) +\frac 1n-\frac{1}{2c_n^2}\,\,.$$
(Hint: apply Taylor's theorem to the function $f(x)=\ln(n+x)$. Observe
that 
$f(0)=\ln(n)$, and use Taylor's theorem with the degree of the
remainder equal to~$2$ to find a formula for $f(1)$.)

\vfill\eject
\smallskip\noindent (c) Rewrite the equation from part (b) in the form
$$\frac 1n +\ln(n)-\ln(n+1)=\frac{1}{2c_n^2}\,.$$
By adding these equations up as $n$ varies, show that 
for any $k\ge1$ there is an equality
$$\left(\sum_{n=1}^k\frac1n\right) -\ln(k+1) =
\sum_{n=1}^k\frac{1}{2c_n^2}\,\,.$$ 

\smallskip\noindent (d) 
Recall from part (b) that $c_n>n$. Using this, conclude from part (c)
that the limit
$$\lim_{k\to\infty}\left[\left(\sum_{n=1}^k\frac1n\right) -\ln(k+1)
\right]$$
exists. (You might want to appeal to the comparison test for the
convergence of series). This value of this limit is called {\sl
Euler's constant}, and is approximately $0.577216$. No one knows
whether or not Euler's constant is a rational number.

\smallskip\noindent (e) {\bf Extra Credit} (10 pts) Give a geometric
interpretation of Euler's constant in terms of areas of region(s) 
in the plane.


\prob6 (20 pts) Let $f(x)$ be the function defined by the formula
$$f(x)=\int_0^{\sin x}e^{t^2}dt\,\,.$$
Compute $f'(x)$. (Hint: let $g(y)=\int_0^ye^{t^2}dt$, and note that
$f(x)=g(\sin x)$.)



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