Math 213 Advanced Calculus Final Exam SOLUTIONS

January 3, 2001 Ali Sinan Sertöz

1) Construct two sequences of real numbers, x_{n} ≥ 0 and y_{n} ≥ 0, such that lim sup_{n→∞}(x_{n}y_{n}) < (lim sup_{n→∞} x_{n})(lim sup_{n→∞} y_{n})

Several such constructions are possible. Take for example

x_{n} = \{ \begin{matrix}1 + \frac{1}{n} & n odd \\ \frac{1}{n} & n even \end{matrix}

and

y_{n} = \{ \begin{matrix}1 + \frac{1}{n} & n even \\ \frac{1}{n} & n odd \end{matrix}

Then x_{n}y_{n} = (1/n)+(1/n^{2}) and converges to zero. Hence lim sup_{n→∞}(x_{n}y_{n}) = 0, lim sup_{n→∞} x_{n} = lim sup_{n→∞} y_{n} = 1.

2) Let f(x) = x tan^{2} x for x ∈ (0, π/2). Calculate (f^{−1})′(π).

See the Inverse Function Theorem on page 96. Using the formula (f^{−1})′(y_{0}) = 1/f′(x_{0}), where y_{0} = f(x_{0}), and noting that here x_{0} = π/3 one easily finds that the answer is (3 + (8/\sqrt{3})π)^{−1}.

3) Show that I_{p} is conditionally convergent for 0 < p ≤ 1, and absolutely convergent for p > 1, where I_{p} = ∫_{1}^{∞} \frac{sin x}{x^{p}}dx.

For the conditionally convergent part look at Example 5 on page 130-131. For the second part use the Comparison Theorem on page 129.

4) Find the \underline{interval} of convergence of the series ∑_{n=1}^{∞} \frac{1 · 3 ⋯ (2n + 1)}{n!}x^{n}.

Ratio test gives |x| < 1/2 for absolute convergence. When |x| = 1/2 the general term of the series does not converge to zero, so the series diverges.

5) Show that in any metric space a compact set is always closed.

One proof is on page 236, Remark 4. This proof depends on Theorem 5.15 on page 235. Reading the two proofs carefully one sees that no property of ℝ^{n} is used other than the existence of a metric, so the proof goes for an arbitrary metric space. I gave another proof in class which started by choosing a point y outside the compact proper set E and considering the open cover ∪_{x∈E}B_{d(x,y)/2}(x). A finite subcover exists. Choosing r > 0 smaller than the radii of the balls in this finite subcover we see that the open ball B_{r}(y) totally lies in the complement of E. Thus E^{c} is open and E is closed.