Mathematics 242, section 1 – Principles of Analysis Final Examination – May 15, 2013

Directions

Do all of the following problems I - VII in the blue exam booklet. There are 200 possible regular points and 10 Extra Credit points. Even if you cannot see how to answer some question completely, for possible partial credit, state a theorem or a result you think applies. You have until 5:30pm.

I. (20) Let A = {x^{2} − 2 : −1 < x < 2} and B = {x : |x − 1| < 4}. Find lub(A ∪ B) and glb(A ∩ B).

II.

A) (10) State the ε, n_{0} definition of convergence for a sequence of real numbers. B) (10) Identify lim_{n→∞} \frac{5n^{2}+1}{n^{2}+n+4}. C) (10) Show that your result in part B is correct using the definition.

III.

A) (15) 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. B) (10) True/False and prove/give a reason: The sequence

x_{n} = \{ \begin{matrix}−1 + \frac{1}{n^{2}} & if n is a prime integer > 1 \\ 1 − \frac{1}{n^{2}} & if n is not prime \end{matrix}

has convergent subsequences. C) (10) True/False and prove/give a reason: The infinite “continued radical”

\sqrt{3 + \sqrt{3 + \sqrt{3 + \sqrt{3 + \sqrt{3 + ⋯}}}}}

represents a finite real number. (Hint: If so, that number would be the limit of a sequence defined by x_{1} = \sqrt{3} and x_{n} = \sqrt{3 + x_{n−1}} for all n ≥ 2.)

IV. A) (15) Let

f(x) = \{ \begin{matrix}x + 3 & if x is a rational number \\ −x^{2} + 3 & if x is an irrational number \end{matrix}

Is f continuous at x = 0? Why or why not? B) (25) State and prove the Intermediate Value Theorem. (You may assume as known

the theorem that if f is continuous at c and x_{n} → c is a sequence contained in the domain of f, then f(x_{n}) → f(c)).

A) (15) Using the limit definition of the derivative, compute f′(c) for f(x) = \frac{1}{(x+3)^{2}} at a

general c ≠ −3. B) (10) What theorem guarantees that

F (x) = ∫_{1}^{x} \frac{1}{(t + 3)^{2}} dt

is differentiable at x = 2? Exactly why does it apply here? What does it say about F′(2)?

C) (10) Show that if f(x) = \frac{e^{x}+e^{−x}}{2} , then for every real k, there exists a solution c of the equation f′(c) = \frac{e^{c}−e^{−c}}{2} = k.

VI. (20) In this question, you may use the summation formulas:

∑_{i=1}^{n} 1 = n ∑_{i=1}^{n} i = \frac{n(n + 1)}{2} ∑_{i=1}^{n} i^{2} = \frac{n(n + 1)(2n + 1)}{6}.

Show that f(x) = x^{2} + x is integrable on [a, b] = [0, 2] by considering upper and lower sums for f and determine the value of ∫_{0}^{2} x^{2} + x dx by using a limit of the upper sums.

VII. A) (5) State the definition of convergence for an infinite series ∑_{n=1}^{∞} a_{n}. B) (15) For which x ∈ R does the power series

∑_{n=1}^{∞} \frac{3^{n}x^{n}}{n7^{n}}

converge? (Apply the Ratio Test and then test the series at the endpoints of your interval separately.)

Extra Credit (10) Prove the result mentioned in part B of question IV above: if f is continuous at c and x_{n} → c is a sequence contained in the domain of f, then f(x_{n}) → f(c).

Have an enjoyable and productive summer!