Mathematics 242 – Principles of Analysis Final Examination May 17, 2011

Directions

Do all of the following problems I - VII in the blue exam booklet. There are 200 possible points. Even if you cannot see how to answer some question completely, for possible partial credit, state a definition, theorem, or result you think applies.

I.

A) (10) State the ϵ, n_{0} definition of convergence for a sequence of real numbers. B) (10) Identify lim_{n→∞} \frac{2n^{2}−2}{n^{2}+n}. C) (20) Show that your result in part B is correct using the definition.

II. (30) Show using mathematical induction that the sequence {x_{n}} defined by x_{1} = 1, and

x_{n} = \sqrt{x_{n−1} + 1}, if n ≥ 2

is bounded above by 2 and monotone increasing. Does this imply that {x_{n}} is convergent? If so, say why and find the limit. If not, say why not.

III. (25) State and prove the Mean Value Theorem. (Prove the special case known as Rolle’s Theorem first, and then deduce the general result from that.)

IV. True/False. For each true statement, give a short proof. For each false statement, give a counterexample or a reason.

A) (10) Let f be differentiable and f′ be continuous on [a, b]. If f′(x) ≠ 0 for all x ∈ [a, b], then f(x) is either increasing on the whole interval [a, b] or decreasing on the whole interval [a, b].

B) (10) If f is uniformly continuous on [a, b], then f is differentiable at every point in

[a, b]. C) (10) Let f : [a, b] → R be continuous and let {x_{n}} be a sequence with x_{n} ∈ [a, b] for

all n. Then the sequence {f(x_{n})} has a convergent subsequence.

V. (25) Show that

f(x) = \{ \begin{matrix}1 & if 0 ≤ x < 1/2 \\ 2 & if 1/2 ≤ x ≤ 1 \end{matrix}

is integrable on [a, b] = [0, 1] by considering upper and lower sums for f. Also, determine the value of ∫_{0}^{1} f.

A) (10) Noting that

lim_{x→1} \frac{ln(x)}{x − 1} = lim_{x→1} \frac{ln(x) − ln(1)}{x − 1},

evaluate the limit. (You may use any calculus facts you need here without proof.) B) (10) Let x = 1 + \frac{1}{n} in the limit from part A, and use that result and the sequential

criterion for continuity applied to f(x) = e^{x} to compute

lim_{n→∞} (1 + \frac{1}{n})^{n} .

VII. A) (5) State the definition of convergence for an infinite series ∑_{n=1}^{∞} a_{n}. B) (5) Does the series ∑_{n=1}^{∞} \frac{2^{n}}{5^{n}} converge? If so, what is the sum of the series? C) (10) Does the series ∑_{n=1}^{∞} ne^{−n} converge? (Use the Integral Test.) D) (10) Use the Ratio Test to determine the x ∈ R for which the series ∑_{n=1}^{∞} \frac{nx^{n}}{2^{n}}

converges absolutely.

Extra Credit (20) Let

f = \{ \begin{matrix}x + x^{2} cos(π/x) & if x ≠ 0 \\ 0 & if x = 0. \end{matrix}

Show that f is differentiable at all x ∈ R, but f′ is not continuous at x = 0.

Have an enjoyable and productive summer!