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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Final Examination}
\centerline{May 17, 2011}
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{\it Directions}
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Do all of the following problems I - VII in the blue exam booklet.  
There are 200 possible points.  {\it Even if you cannot see how
to answer some question completely, for possible partial credit, 
state a definition, theorem, or result you think applies.}
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I.
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\item{A)} (10) State the $\epsilon$, $n_0$ definition of convergence for 
a sequence of real numbers.
\item{B)} (10) Identify $\lim_{n\to\infty} {2n^2 - 2\over n^2 + n}$.
\item{C)} (20) Show that your result in part B is correct using the 
definition.
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II.  (30) Show using mathematical induction that the 
sequence $\{x_n\}$ defined by $x_1 = 1$, and 
$$x_n = \sqrt{x_{n-1} + 1}, \qquad  {\rm if}\ n \ge 2$$
is bounded above by $2$ and monotone increasing.  Does this
imply that $\{x_n\}$ is convergent?  If so, say why and find
the limit.  If not, say why not.   
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III.  (25) State and prove the Mean Value Theorem.
(Prove the special case known as Rolle's Theorem first, and 
then deduce the general result from that.)
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IV.  True/False.  For each true statement, give a 
short proof.  For each false statement, give a counterexample or a reason.
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\item{A)} (10) Let $f$ be differentiable and $f'$
be continuous on $[a,b]$.  If $f'(x) \ne 0$ for all $x \in [a,b]$, 
then $f(x)$ is either increasing on the whole interval $[a,b]$
or decreasing on the whole interval $[a,b]$.
\item{B)} (10) If $f$ is uniformly continuous on $[a,b]$, then 
$f$ is differentiable at every point in $[a,b]$.  
\item{C)} (10) Let $f : [a,b] \to {\bf R}$ be continuous and let
$\{x_n\}$ be a sequence with $x_n \in [a,b]$ for all $n$.  Then
the sequence $\{f(x_n)\}$ has a convergent subsequence.  
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V.  (25) 
Show that 
$$f(x) = \cases{1 & if $0 \le x < 1/2$\cr
                2 & if ${1/2} \le x \le 1$\cr}$$
is integrable on $[a,b] = [0,1]$ by 
considering upper and lower sums for $f$.  Also, determine the value of 
$\int_0^1 f$.  
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VI.  
\item{A)} (10) Noting that 
$$\lim_{x\to 1} {\ln(x)\over x - 1} = \lim_{x\to 1} {\ln(x) - \ln(1) \over x - 1},$$
evaluate the limit.  (You may use any calculus facts you need here without 
proof.)
\item{B)}  (10)  Let $x = 1 + {1\over n}$ in the limit from part A, and use 
that result and the sequential criterion for continuity 
applied to $f(x) = e^x$ to compute 
$$\lim_{n\to \infty} \left(1 + {1\over n}\right)^n.$$
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VII.  
\item{A)} (5) State the definition of convergence for an infinite
series $\sum_{n=1}^\infty a_n$. 
\item{B)} (5) Does the series $\sum_{n=1}^\infty {2^n\over 5^n}$
converge?  If so, what is the sum of the series?
\item{C)} (10) Does the series $\sum_{n=1}^\infty {n e^{-n}}$
converge?  (Use the Integral Test.)
\item{D)} (10) Use the Ratio Test to determine the $x \in {\bf R}$ for which the
series $\sum_{n=1}^\infty {n x^n \over 2^n}$
converges absolutely.
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{\it Extra Credit} (20) Let 
$$f = \cases{x + x^2 \cos(\pi/x) & if $x \ne 0$\cr
                    0 & if $x = 0$.\cr}$$
Show that $f$ is differentiable at all $x \in {\bf R}$, but $f'$ 
is not continuous at $x = 0$.   
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\centerline{\it Have an enjoyable and productive summer!}
\bye              
