Comprehensive Examination Advanced Calculus January 1999

A passing mark will be given if 5 problems are completely solved.

1. Determine whether each of the following statements is true or false. If it is true, prove it; if it is false, give a counterexample. In each case (a_{n})_{n=1}^{∞} is a sequence of real numbers.

(a) The series ∑_{n=1}^{∞} a_{n} converges if ∑_{n=1}^{∞} a_{n}^{2} converges.

(b) The series ∑_{n=1}^{∞} a_{n} converges if a_{n} ≥ 0 for all n and ∑_{n=1}^{∞} \sqrt{a_{n}} converges.

2. Suppose 0 < p < 1. Prove that t^{p} + 1 ≥ (1 + t)^{p} for all t ≥ 0.

3. Let

f(x, y) = \begin{cases}\frac{xy^{2}}{x^{2} + y^{2}} & (x, y) ≠ (0, 0), \\ 0, & (x, y) = (0, 0). \end{cases}

Prove or disprove f is differentiable at (0, 0).

4. Suppose that (f_{n})_{n=1}^{∞} is a sequence of continuous real-valued functions on a metric space (X, d).

(a) Define (f_{n})_{n=1}^{∞} converges uniformly to f : X → ℝ.

(b) Prove that if X is the real line and (f_{n})_{n=1}^{∞} converges uniformly to f, then f is continuous.

5. Let X, Y and Z be metric spaces and assume that Y is compact. Suppose that f : X → Y, g : Y → Z and that g is continuous and one-to-one. Define h(x) = g(f(x)) for all x ∈ X. Prove that f is uniformly continuous if h is uniformly continuous.

6. Define

f(x) = \begin{cases}sin(\frac{1}{x}) & 0 < x ≤ 1, \\ 0 & x = 0, \end{cases} and f_{n}(x) = [f(x)]^{n} for n = 1, 2, ….

(a) For each x ∈ [0, 1] determine whether or not lim_{n→∞} f_{n}(x) exists and carefully justify your results.

(b) Compute lim_{n→∞} ∫ 1_{\frac{1}{π}} f_{n}(x) dx and justify your computation.

7. Let f : [a, b] → ℝ^{n} for some n > 1.

(a) Define f is rectifiable.

(b) Prove f is not rectifiable if

f(x) = \begin{cases}(x, x sin(\frac{1}{x})) & x ∈ (0, 1], \\ (0, 0) & x = 0. \end{cases}