Math 312, Intro. to Real Analysis: Final Exam: Solutions

Stephen G. Simpson

Friday, May 8, 2009

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}. True.

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

(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]. True.

(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). False.

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

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.

Solution. The only axiom that fails for ℚ is the Completeness Axiom. For example, the set {x ∈ ℚ | x^{2} < 2} is bounded but has no least upper bound in ℚ.

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.

Solution. Given ϵ > 0 we can find N such that |s_{n} − s| < ϵ for all n > N.

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

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

Solution. Clearly |π/\sqrt{n − 100π}| < ϵ whenever n > 100π + π^{2}/ϵ^{2}. Thus we may take N = 100π + π^{2}/ϵ^{2}.

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

Solution. The sum of this geometrical series is

\frac{3}{2} · (1 − \frac{1}{2} + \frac{1}{4} − ⋯) = \frac{3}{2} · \frac{1}{1 − (−\frac{1}{2})} = 1.

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

Solution. 0, 1, 0, 1, 0, 1, . . ..

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

Solution. 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, . . ..

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

Solution. 0, 1, 0, \frac{1}{2}, 1, 0, \frac{1}{3}, \frac{2}{3}, 1, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, . . ..

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].

Solution. Let f_{n}(0) = 0, f_{n}(1/n) = 1, f_{n}(2/n) = f_{n}(1) = 0, and let f_{n}(x) be linear on the intervals [0, 1/n], [1/n, 2/n] and [2/n, 1].

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

Hint: Use the Triangle Inequality.

Solution. We know that ∑ a_{k} is convergent if and only if the Cauchy criterion holds: given ϵ > 0 we can find N such that |a_{m}+a_{m+1}+⋯+a_{n}| < ϵ whenever n > m > N. The Triangle Inequality tells us that

|a_{m} + a_{m+1} + ⋯ + a_{n}| ≤ |a_{m}| + |a_{m+1}| + ⋯ + |a_{n}|.

Thus, ∑ a_{k} satisfies the Cauchy criterion whenever ∑ |a_{k}| does.

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}}

Solution. Convergent geometric series.

(b) ∑ \frac{(−1)^{n}}{\sqrt{n}}

Solution. Convergent by the Alternating Series Test.

(c) ∑(−1)^{n} \frac{n}{100n + 1000}

Solution. Divergent by the nth term test, because \frac{n}{100n + 1000} →

\frac{1}{100} ≠ 0.

(d) ∑ (\frac{7n}{8n + 1})^{n}

Solution. Convergent by the Root Test, because \frac{7n}{8n + 1} → \frac{7}{8} < 1.

(e) ∑ \frac{1}{n log n}

Solution. Divergent by the Integral Test.

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

Solution. Uniformly continuous on (−∞, ∞).

(b) e^{x}

Solution. Uniformly continuous on any bounded interval; continuous but not uniformly continuous on [a, ∞) for any a.

(c) |x − \frac{1}{2}| + |x − 3|

Solution. Uniformly continuous on (−∞, ∞).

(d) \frac{1}{1 − x}

Solution. Undefined at x = 1, hence not continuous on [0, 1]. Continuous but not uniformly continuous on (0, 1). Uniformly continuous on [2, ∞).

(e) ∑_{n=0}^{∞} x^{n}

Solution. Undefined at x = 1, hence not continuous on [0, 1]. Continuous but not uniformly continuous on (0, 1). Undefined (divergent) for x > 1.

10. (10 points)

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

Solution. For all x ∈ [−0.99, 0.99] we have |x^{n}| ≤ 0.99^{n}, and 0.99^{n} → 0 independently of x.

(b) Does x^{n} → 0 uniformly on the interval (−1, 1)? Justify your answer. Solution. No, because for any n we have 0^{n} = 0 and

1^{n} = 1, hence we can find x such that 0 < x < 1 and x^{n} = 1/2.

11. (10 points)

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

Solution. We have

lim_{n→∞} ∫_{−1}^{1} sin^{n} x dx = ∫_{−1}^{1} lim_{n→∞} sin^{n} x dx = ∫_{−1}^{1} 0 dx = 0.

(b) Justify your calculation for part (a) by stating an appropriate property of the functions sin^{n} x and an applicable theorem.

Solution. We have sin^{n} x → 0 uniformly on [−1, 1]. The applicable theorem says: if f_{n} → f uniformly on [a, b] and f_{n} and f are continuous, then lim ∫_{a}^{b} f_{n} = ∫_{a}^{b} f.

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

Solution. Diverges whenever cos x = ±1, i.e., x = ±nπ. Converges everywhere else, because then | cos x| < 1.

(b) ∑_{n=1}^{∞} \frac{1}{nx^{n}}

Solution. Diverges for −1 < x ≤ 1, converges everywhere else.

(c) ∑_{n=0}^{∞}(2 + \sqrt{n})x^{n}

Solution. Converges for −1 < x < 1, diverges everywhere else.

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

Solution. Converges for −1/10 ≤ x ≤ 1/10, diverges everywhere else.

(e) ∑_{n=0}^{∞} n(x − 2)^{n}

Solution. Converges for 1 < x < 3, diverges everywhere else.

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)

Solution. We have

\frac{x^{2} + 1}{x − 1} = \frac{(x + 1)(x − 1) + 2}{x − 1} = 1+x−\frac{2}{1 − x} = 1+x−2(1+x+x^{2}+⋯).

Thus a_{0} = a_{1} = −1, and a_{k} = −2 for all k ≥ 2.

(b) Over what interval is the above equation (1) valid? Solution. The equation is valid over the interval (−1, 1).