Math 312, Intro. to Real Analysis: Final Exam

Stephen G. Simpson

Friday, May 8, 2009

There are problems worth a total of 150 points. Please be sure to attempt all problems.

1. True or false (3 points each).

(a) For all sequences of real numbers (s_{n}) we have lim inf s_{n} ≤ lim sup s_{n}.

(b) Every bounded sequence of real numbers has at least one subsequential limit.

(c) If the functions f_{n} are continuous on [0, 1] and converge uniformly to the function f on [0, 1], then f is uniformly continuous on [0, 1].

(d) If the radius of convergence of a power series ∑ a_{k}x^{k} is R, and if 0 < R < ∞, then the series ∑ a_{k}x^{k} converges uniformly on (−R, R).

(e) The integral of the limit is equal to the limit of the integrals.

2. (10 points) The real number system ℝ has been characterized in terms of Axioms A1–A4, M1–M4, DL, O1–O5, and the Completeness Axiom. Which of these axioms fail for the rational number system ℚ? Give one or more examples illustrating your answers.

3. (10 points)

(a) State the formal definition of what it means for a sequence of real numbers (s_{n}) to converge to a limit s.

(b) In terms of your definition from part (a), prove directly that

lim_{n→∞} \frac{π}{\sqrt{n − 100π}} = 0.

4. (7 points) Calculate lim_{n→∞} (\frac{3}{2} − \frac{3}{4} + \frac{3}{8} − \frac{3}{16} + ⋯ + (−1)^{n} \frac{3}{2^{n}} ).

5. (a) (2 points) Give an example of a bounded sequence of real numbers with exactly two subsequential limits.

(b) (2 points) Give an example of a bounded sequence of real numbers with exactly five subsequential limits.

(c) (4 points) Give an example of a bounded sequence of real numbers with infinitely many subsequential limits.

6. (10 points) Give an example of a sequence of continuous functions on [0, 1] such that f_{n} → 0 pointwise but not uniformly on [0, 1].

7. (15 points) Prove that if ∑ |a_{k}| is convergent then ∑ a_{k} is convergent.

Hint: Use the Triangle Inequality.

8. (3 points each) For each of the following series, tell whether the series is convergent or divergent. State which convergence/divergence test you are using, and show any needed calculations.

(a) ∑ \frac{1}{2^{n}}
(b) ∑ \frac{(−1)^{n}}{\sqrt{n}}

(c) ∑(−1)^{n} \frac{n}{100n + 1000}

(d) ∑ (\frac{7n}{8n + 1})^{n}
(e) ∑ \frac{1}{n log n}

9. (15 points) For each of the following functions, say whether the function is continuous and/or uniformly continuous on each of the three intervals

[0, 1],  (0, 1),  (2, ∞).

You are not required to justify or prove your answers.

(a) sin x

(b) e^{x}

(c) |x − \frac{1}{2}| + |x − 3|

(d) \frac{1}{1 − x}

(e) ∑_{n=0}^{∞} x^{n}

10. (10 points)

(a) Prove directly that x^{n} → 0 uniformly on the interval [−0.99, 0.99].

(b) Does x^{n} → 0 uniformly on the interval (−1, 1)? Justify your answer.

11. (10 points)

(a) Calculate lim_{n→∞} ∫_{−1}^{1} sin^{n} x dx.

(b) Justify your calculation for part (a) by stating an appropriate prop-

erty of the functions sin^{n} x and an applicable theorem.

12. (15 points) For each of the following series, determine the set of all x such that the series converges at x. Show any needed calculations.

(a) ∑_{n=0}^{∞} 2 cos^{n} x

(b) ∑_{n=1}^{∞} \frac{1}{nx^{n}}

(c) ∑_{n=0}^{∞}(2 + \sqrt{n})x^{n}

(d) ∑_{n=0}^{∞} \frac{(10x)^{n}}{n^{2} + 1}

(e) ∑_{n=0}^{∞} n(x − 2)^{n}

13. (10 points)

(a) Determine the coefficients a_{k} for k = 0, 1, 2, … such that

∑_{k=0}^{∞} a_{k}x^{k} = \frac{x^{2} + 1}{x − 1} .  (1)

(b) Over what interval is the above equation (1) valid?