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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Practice Final Examination}
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\noindent
I.  Let $A = \{x^2 - 2 : -1 < x < 3\}$ and $B = \{x : |x - 1| < 4\}$.
Find $\sup(A \cup B)$ and $\inf(A \cap B)$.
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\noindent
II.
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\item{A)} State the $\epsilon$, $N$ definition of convergence for 
a sequence of real numbers.
\item{B)} Identify $\lim_{n\to\infty} {5n+1\over n + 4}$.
\item{C)} Show that your result in part B is correct using the 
definition.
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\noindent
III.  
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\item{A)}  Show that if $x_n$ is a monotone increasing sequence
of real numbers that is bounded above, then $x_n$ converges
to some real number.
\item{B)}  What general theorem could you use to deduce
that the sequence $x_n = (-1)^n + {(-1)^{n+1}\over n^2}$
has convergent subsequences?
\item{C)}  Let $x_n = (-1)^n + {(-1)^{n+1}\over n^2}$.  Find
a monotone increasing subsequence of $(x_n)$ and identify the limit.
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\noindent
IV.  
\item{A)} Show that if $f(x)$ is continuous at $x = c$, 
and $x_n$ is any sequence satisfying 
$\lim_{n\to\infty} x_n = c$, then 
$$\lim_{n\to\infty} f(x_n) = f(\lim_{n\to\infty} x_n) = f(c).$$  
\item{B)} Let 
$$f(x) = \cases{ x & if $x$ is a rational number\cr
                -x & if $x$ is an irrational number\cr}$$
Is $f$ continuous at $x = 1$?  Why or why not? What about at $x = 0$?
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\noindent
V. 
\item{A)} State and prove the Intermediate Value Theorem.
\item{B)} Show that for every real $y_0$, there exists a solution $x$
of the equation ${e^x - e^{-x}\over 2} = y_0$
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\noindent
VI.  (15)  Using the definition, compute $f'(x)$ for 
$f(x) = {1\over (x+3)^2}$.
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\noindent
VII.  
\item{A)}  (20) In this part you may use the summation
formulas:
$$\sum_{i=1}^n 1 = n \quad \sum_{i=1}^n i = {n^2 + n \over 2} \quad
\sum_{i=1}^n i^2 = {2n^3 + 3n^2 + n \over 6}.$$ 
Show that $f(x) = x^2 + 5$ is integrable on $[a,b] = [0,1]$ by 
considering upper and lower sums for $f$ and determine the value of
$\int_0^1 x^2 + 5\, dx$.
\item{B)} (15) State and prove the Fundamental Theorem of Calculus, ``part 1.''
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\noindent
VIII.  
\item{A)} State the definition of convergence for an infinite
series $\sum_{n=1}^\infty a_n$. 
\item{B)} Suppose that in an infinite series $\sum_{n=1}^\infty a_n$,
$a_n < 0$ for all $n$.  Show that if the partial sum $s_N$ satisfies 
$s_N > B$ for all $N$, then $\sum_{n=1}^\infty a_n$ converges. (Hint:  
What kind of sequence is $\{s_N\}$?)
\item{C)} 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{D)} Use the Integral Test to determine whether
$$\sum_{n=2}^\infty {1\over n (\ln n)^2}$$
converges.
\item{E)} Use the Comparison Test to determine whether the
series 
$$\sum_{n=1}^\infty {n^2 \over n^{3.9} + 1}$$
converges absolutely.
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\noindent
IX.  
\item{A)} 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)}  Let $h = {1\over n}$ in the limit from part A, and use 
that result and the theorem from question IV A on this exam with
$f(x) = e^x$ to compute 
$$\lim_{n\to \infty} \left(1 + {1\over n}\right)^n.$$
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