Mathematics 242 – Principles of Analysis Midterm Exam 2 – October 29, 2004

Directions: Do all work in the blue exam booklet. There are 100 possible regular points (distributed as indicated in the questions below), and 10 Extra Credit points.

I. Let A = {x ∈ R : 1 < x^{2} < 4} and let B be the interval [−1, 3/2).

A) (10) What is the set S = A ∩ B? B) (10) Find sup(S) and inf(S) for the set S from part A.

II. A) (10) State the Completeness (Least Upper Bound) Axiom for the real number system. B) (20) Prove that if x_{n} is a monotone increasing sequence of real numbers that is bounded

above, then lim_{n→∞} x_{n} = L for some L ∈ R. C) (5) Suppose that x_{n} is a sequence of negative numbers and x_{n+1}/x_{n} ≤ 1 for all n ≥ 1.

Show that x_{n} converges to some L ∈ R.

III. A) (15) Identify lim_{n→∞} x_{n} for the sequence

x_{n} = \frac{4n}{7n + 5},

and prove that x_{n} converges to your limit using the ε - N definition of convergence. B) (10) Use the Limit Theorems for sequences to determine

lim_{n→∞} \frac{4n^{2} + 3n + 1}{27n^{3} + 3}

IV. Both parts of this question concern the sequence

x_{n} = cos (\frac{π\sqrt{n}}{2}) + \frac{(−1)^{n}}{n}

A) (10) Show that x_{n} is bounded. B) (10) Find two subsequences of x_{n} converging to two different limits. What does this

say about lim_{n→∞} x_{n}?

Extra Credit. (10) Assume that x_{n} is a sequence that converges to L as n → ∞. Construct a new sequence y_{n} from the terms of x_{n} by y_{n} = \frac{x_{1}+x_{2}+···+x_{n}}{n} Show that y_{n} also converges to L as L → ∞. (Hint: it may help to note that if x_{n} → L, then there is some M such that |x_{n} − L| ≤ M for all n ≥ 1.)