Mathematics 242 – Principles of Analysis Makeup Exam 2 – April 4, 2014

Directions

Do all work in the blue exam booklet. There are 100 possible regular points and 10 Extra Credit Points.

I. A) (20) State and prove the Monotone Convergence Theorem for sequences. (You may

give the proof in the case that the sequence is monotone increasing.)

B) (10) Let {x_{n}} be the sequence defined by x_{1} = 5 and

x_{n+1} = \frac{12 + x_{n}^{2}}{15}

for all n ≥ 1. Does this sequence converge? Why? If it does, what is the limit?

II. A) (15) Show using the ε, n_{0} definition that

lim_{n→∞} \frac{4n + 1}{7n + 3} = \frac{4}{7}.

B) (15) Show using the ε, δ definition that

f(x) = \frac{4x + 1}{7x + 3}

is continuous at x = 0.

III. (10) Suppose {x_{n}} is a sequence such that |x_{n} − 10| < 30 for all n ≥ 1. Show that there exists some number a ∈ [−20, 40] and a subsequence {x_{n_{k}}} such that x_{n_{k}} → a. State any “big theorems” you are using.

IV. Give an example, or give a reason why there can be no such examples:

A) (10) A function f and an a in the domain of f such that lim_{x→a^{−}} f(x) does not exist, but lim_{x→a^{+}} f(x) does exist.

B) (10) A sequence x_{n} such that x_{n} → −4, but for all n_{0} ∈ N there exist n ≥ n_{0} with x_{n} > 0.

C) (10) A bounded function that is not continuous at any c in its domain.

Extra Credit. (10) Assume that {x_{n}} is a sequence that converges to a. Construct a new sequence {y_{n}} by making y_{n} the average of the terms x_{n}, x_{n+1}, x_{n+2}, x_{n+3}, x_{n+4} in the original sequence. So for instance,

y_{1} = \frac{x_{1} + x_{2} + x_{3} + x_{4} + x_{5}}{5} , y_{2} = \frac{x_{2} + x_{3} + x_{4} + x_{5} + x_{6}}{5} ,

and so on. Show that {y_{n}} also converges to a.