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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Final Examination}
\centerline{December 14, 2005}
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{\it Directions}
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Do all of the following problems I - IX 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 theorem or a result you think applies.}
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I.
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\item{A)} (10) State the $\epsilon$ -- $N$ definition of convergence for 
a sequence of real numbers.
\item{B)} (5) Identify $\lim_{n\to\infty} {5n^2+3n+1\over n^2 + 4n}$.
\item{C)} (10) Show that your result in part B is correct using the 
definition.
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II.  (15) Show using mathematical induction that the 
sequence $(x_n)$ defined by $x_1 = 1$, and 
$x_n = \sqrt{x_{n-1} + 1}$ for $n \ge 2$,
is bounded above by $2$ and monotone increasing.  Identify the limit
$\lim_{n\to\infty} x_n$.
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III.  (20) State and prove the Bolzano-Weierstrass theorem.
(You may assume without proof the Nested Interval Property
of ${\bf R}$.)
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IV.  
\item{A)} (10) State the definition of convergence for an infinite
series $\sum_{n=1}^\infty a_n$. 
\item{B)} (10) Does the series $\sum_{n=1}^\infty {5\cdot 3^n\over 7^n}$
converge?  If so, what is the sum of the series?
\item{C)} (10) Does the series $\sum_{n=1}^\infty {(-1)^n\over n + \ln(n)}$
converge absolutely, converge conditionally, or diverge?
\item{D)} (10) Use the Ratio Test to determine whether the
series $\sum_{n=1}^\infty {(-1)^n n^2 \over 2^n}$
converges absolutely.
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V.  
Let 
$$f(x) = \cases{ x^2 - 1 & if $x \in {\bf Q}$\cr
                2x - 2 & if $x \notin {\bf Q}$\cr}$$
\item{A)} (10) Is $f$ continuous at $x = 0$?  Prove your assertion.
\item{B)} (10) Is $f$ continuous at $x = 1$?  Prove your assertion.

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VI.  True/False.  For each true statement, give a 
short proof.  For each false statement, give a counterexample.
\item{A)} (10) If $f$ is continuous on $[a,b]$ and $f(x) \ne 0$
for all $x\in [a,b]$, then $f(x)$ is either always positive or
always negative on $[a,b]$.
\item{B)} (10)  If $f$ is continuous on $[a,b]$ and $f(x) > 0$
for all $x\in [a,b]$, then there exists an $m > 0$
such that $f(x) \ge m$ for all $x$ in $[a,b]$.
\item{C)} (10) If $f$ is continuous on $(a,b)$, then
the range $f((a,b))$ is an open (but not necessarily 
bounded) interval.

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VII.  (15)  Using the definition, compute $f'(0)$ for 
$$f(x) = \cases{x^3\sin\left({1\over x}\right) + 3x & if $x\ne 0$ \cr
                 0  & if $x = 0$ \cr}.$$
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VIII.  (20) In this part you may use the summation
formulas:
$$\sum_{i=1}^n 1 = n \quad \sum_{i=1}^n i = {n(n+1)\over 2} \quad
\sum_{i=1}^n i^2 = {n(n+1)(2n+1) \over 6}.$$ 
Show that $f(x) = x^2 + 5$ is integrable on $[a,b] = [1,3]$ by 
considering upper and lower sums for $f$ and determine the value of
$\int_1^3 x^2 + 5\, dx$ using the upper sum.
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IX.  
\item{A)} (10) Noting that 
$$\lim_{h\to 0} {\ln(1+h)\over h} = \lim_{h\to 0} {\ln(1+h) - \ln(1) \over h},$$
evaluate the limit.  (You may use any calculus facts you need here without 
proof.)
\item{B)}  (5)  Let $h = {1\over n}$ 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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{\it Extra Credit} (20) (The Integral Test for series)
Let $f(x)$ be a continuous, non-negative, monotone decreasing
function for all $x > 0$.
Show that if $\lim_{B\to\infty} \int_1^B f(x)\ dx$
exists (and is finite), then the series 
$$\sum_{n=1}^\infty f(n) = f(1) + \sum_{n=2}^\infty f(n)$$
converges.  Use this to show $\sum_{n=1}^\infty {1\over n^\alpha}$
converges for all $\alpha > 1$.

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\centerline{\it Best wishes for a peaceful and joyous holiday season!}
\bye              
