Mathematics 242 – Principles of Analysis Final Examination December 14, 2005

Directions

Do all of the following problems I - IX 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 theorem or a result you think applies.

I.

A) (10) State the ϵ – N definition of convergence for a sequence of real numbers. B) (5) Identify lim_{n→∞} \frac{5n^{2}+3n+1}{n^{2}+4n}. C) (10) Show that your result in part B is correct using the definition.

II. (15) Show using mathematical induction that the sequence (x_{n}) defined by x_{1} = 1, and x_{n} = \sqrt{x_{n−1} + 1} for n ≥ 2, is bounded above by 2 and monotone increasing. Identify the limit lim_{n→∞} x_{n}.

III. (20) State and prove the Bolzano-Weierstrass theorem. (You may assume without proof the Nested Interval Property of R.)

IV. A) (10) State the definition of convergence for an infinite series ∑_{n=1}^{∞} a_{n}. B) (10) Does the series ∑_{n=1}^{∞} \frac{5·3^{n}}{7^{n}} converge? If so, what is the sum of the series? C) (10) Does the series ∑_{n=1}^{∞} \frac{(−1)^{n}}{n+ln(n)} converge absolutely, converge conditionally, or di-

verge? D) (10) Use the Ratio Test to determine whether the series ∑_{n=1}^{∞} \frac{(−1)^{n}n^{2}}{2^{n}} converges

absolutely.

V. Let

f(x) = \{ \begin{matrix}x^{2} − 1 & if x ∈ Q \\ 2x − 2 & if x ∉ Q \end{matrix}

A) (10) Is f continuous at x = 0? Prove your assertion. B) (10) Is f continuous at x = 1? Prove your assertion.

VI. True/False. For each true statement, give a short proof. For each false statement, give a counterexample. A) (10) If f is continuous on [a, b] and f(x) ≠ 0 for all x ∈ [a, b], then f(x) is either

always positive or always negative on [a, b]. B) (10) If f is continuous on [a, b] and f(x) > 0 for all x ∈ [a, b], then there exists an

m > 0 such that f(x) ≥ m for all x in [a, b].

C) (10) If f is continuous on (a, b), then the range f((a, b)) is an open (but not necessarily bounded) interval.

VII. (15) Using the definition, compute f′(0) for

f(x) = \{ \begin{matrix}x^{3} sin (\frac{1}{x}) + 3x & if x ≠ 0 \\ 0 & if x = 0 \end{matrix} .

VIII. (20) In this part 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} +5 is integrable on [a, b] = [1, 3] by considering upper and lower sums for f and determine the value of ∫_{1}^{3} x^{2} + 5 dx using the upper sum.

IX. A) (10) Noting that

lim_{h→0} \frac{ln(1 + h)}{h} = lim_{h→0} \frac{ln(1 + h) − ln(1)}{h},

evaluate the limit. (You may use any calculus facts you need here without proof.) B) (5) Let h = \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} .

Extra Credit (20) (The Integral Test for series) Let f(x) be a continuous, non-negative, monotone decreasing function for all x > 0. Show that if lim_{B→∞} ∫_{1}^{B} f(x) dx exists (and is finite), then the series

∑_{n=1}^{∞} f(n) = f(1) + ∑_{n=2}^{∞} f(n)

converges. Use this to show ∑_{n=1}^{∞} \frac{1}{n^{α}} converges for all α > 1.

Best wishes for a peaceful and joyous holiday season!