Comprehensive Exam Advanced Calculus August 2000

A passing score will consist of five problems completed correctly. Please show all work.

1. Suppose that (a_{n}) is a nondecreasing sequence of positive real numbers and there exists a positive integer k such that

lim_{m→∞} \frac{a_{k·m}}{k · m} = 0.

Prove that lim_{m→∞} \frac{a_{m}}{m} = 0.

2. Find the interval of convergence including behavior at the endpoints of the series

∑_{n=1}^{∞} \frac{1}{ne^{n} + n^{2}} (x − 2)^{n}.

3. Define

f(x) = ∑_{n=1}^{∞} \frac{1}{x^{n} + x^{−n}}

wherever the series converges.

(a) Determine the domain of f.

(b) Determine the set of points where f is continuous.

4. Suppose that f : ℝ^{2} → ℝ.

(a) Define f is differentiable at (x_{0}, y_{0}).

(b) If f is defined by

f(x, y) = \begin{cases}e^{−1/(x^{2}+y^{2})} & if (x, y) ≠ (0, 0), \\ 0 & if (x, y) = (0, 0), \end{cases}

Prove that f is differentiable at (0, 0).

5. Let

u(x, y) = \frac{x + y}{x − y} and v(x, y) = \frac{xy}{(x − y)^{2}} .

(a) Compute the Jacobian of the map T (x, y) = (u(x, y), v(x, y)).

(b) Show that there exists a function F : ℝ^{2} → ℝ such that F (u, v) = 0.

6. Let S = \{(\frac{1}{n}, \frac{1}{n}) : n = 1, 2, 3, … \} as a subset of ℝ^{2} with its usual topology. Show that S is not compact by giving an open cover of S with no finite subcover.

7. (a) Define (using sums) what it means for a real-valued function f on an interval [a, b] to be Riemann integrable.

(b) Using the definition prove that the function f defined by

f(x) = \begin{cases}n & if x = \frac{n−1}{n} for some n ∈ ℕ. \\ 0 & otherwise. \end{cases}

is or is not Riemann integrable on [0, 1].