Comprehensive Examination Advanced Calculus June 1998

This exam consists of seven problems to be completed in two hours. A passing score will consist of five problems completed correctly. Please show all work.

1. Let {a_{n}}, {b_{n}} be bounded sequences of real numbers.

(a) Prove that lim sup_{n→∞}(a_{n} + b_{n}) ≤ lim sup_{n→∞} a_{n} + lim sup_{n→∞} b_{n}.

(b) Either prove that equality holds in part (a) or give an example to show that strict inequality can occur in part (a).

2. Let X be a metric space.

(a) Define the term compact in terms of open covers.

(b) Using the definition in the previous part prove that if C is a compact subset of X

and S ⊆ C is infinite then S has a limit point in C.

3. Let f : [0, ∞) → R be a continuous function that is differentiable on (0, ∞). Suppose that f(0) > 0 and f′(x) < 0 for all x > 0. Prove that the equation f(x) = x has one and only one solution.

4. Determine the radius of convergence and the convergence at the endpoints of the interval of convergence for the power series ∑_{n=0}^{∞} a_{n}x^{n}, where a_{n} = (sin \frac{nπ}{3})^{n}.

5. Show that the series ∑_{n=1}^{∞} \frac{x}{n^{2} − x^{2}} converges uniformly on any closed interval [a, b] that

does not contain any of the points ±1, ±2, . . ..

6. Let f : R^{n} → R.

(a) Define the phrase “f is differentiable at (0, 0, …, 0).”

(b) Prove directly from the definition that f : R^{2} → R defined by f(x, y) = x^{2} + y is differentiable at (0, 0).

7. Let f : ℝ^{2} → ℝ be differentiable and define g : S → ℝ by

g(x, y, z) = x^{3}f (\frac{y}{x}, \frac{z}{x}) ,

where S = {(x, y, z) ∈ ℝ^{3} | x ≠ 0}. Show that

x\frac{∂g}{∂x} + y \frac{∂g}{∂y} + z \frac{∂g}{∂z} = 3g.