Mathematics 242 – Principles of Analysis Final Examination December 15, 2000

Directions

Do all work in the blue exam booklet. There are 200 possible regular points and 20 possible Extra Credit points (see problem VII below). I. Let A = {sin(x) : x ∈ [0, 3π/4]} and let B = {x : 1 < x^{2} < 4}. A) (10) What is the set A ∪ B? B) (10) What is the least upper bound of the set C = {|x − 2| : x ∈ A}? II. A) (10) State the ϵ - n_{0} definition for convergence of a sequence. B) (20) Identify L = lim_{n→∞} x_{n} for the sequence

x_{n} = \frac{3n^{2} + n}{n^{2} + 1}

and prove using the definition that lim_{n→∞} x_{n} = L. III. Let x_{n} = sin(n) and y_{n} = cos(n). In this problem you may use without proof any facts about sequences, subsequences, etc. that we proved in class. A) (10) Show that there exists an index sequence of positive integers n_{k} such that x_{n_{k}} =

sin(n_{k}) converges. B) (10) Does there exist an index sequence n_{k} of positive integers such that both x_{n_{k}} and

y_{n_{k}} converge? Prove your assertion. IV. A) (10) Give the ϵ - δ definition for the statement lim_{x→c} f(x) = L. B) (20) Identify the limit

L = lim_{x→0} x^{2} sin(1/x)

and prove using the definition that lim_{x→0} x sin(1/x) = L. V. A) (10) State the Intermediate Value Theorem. B) (10) Let f(x) = a_{1} + a_{2} sin(x), where a_{1}, a_{2} are constants with |a_{1}| < a_{2}. Show that

there exists some c ∈ [−π/2, π/2] such that f(c) = 0. VI. (30) State and prove both parts of the Fundamental Theorem of Calculus.

VII. True - False. For each true statement, give a short proof or reason. For each false statement give a reason or a counterexample. Do any five of the following. If you submit solutions for more than five parts, correct answers will be counted as Extra Credit. In any parts that refer to f(x), use

f(x) = \{ \begin{matrix}x^{2} & if x ∈ [0, 1) \\ \frac{1}{2} & if x ∈ [1, 2] \end{matrix}

A) (10) f(x) is differentiable at x = 1. B) (10) Let ∑_{n=1}^{∞} a_{n} be an infinite series with positive terms. If the partial sums s_{N} are

bounded above, then ∑_{n=1}^{∞} a_{n} converges. C) (10) The conclusion of the Mean Value Theorem holds for f(x) on the interval [0, 2]. D) (10) The infinite series

∑_{n=1}^{∞} \frac{(−1)^{n}}{n^{3}}

converges conditionally. E) (10) f(x) is integrable on [0, 2]. (You may use without proof any general theorems

about integrability.) F) (10) The infinite series

∑_{n=1}^{∞} \frac{x^{n}}{n2^{n}}

converges for all x ∈ R. G) (10) If p > 1, then

\frac{1}{p − 1} ≤ ∑_{n=1}^{∞} \frac{1}{n^{p}} ≤ \frac{p}{p − 1}

Have a wonderful holiday season!