Advanced Calculus Comprehensive Exam January 2000

Four problems completely solved will constitute a pass.

(1) Suppose that (X, d) is a complete metric space and (x_{n}) is a sequence in X such that d(x_{n}, x_{n+1}) < \frac{1}{2^{n}} for n = 1, 2, 3, . . .. Prove that (x_{n}) converges.

(2) Consider f_{n}(x) = nx exp(−nx).

(a) Compute lim_{n→∞} f_{n}(x).

(b) Prove f_{n} converges uniformly on [δ, ∞) for every δ > 0.

(c) Prove or disprove: f_{n} converges uniformly on [0, ∞).

(3) Let a_{n} be a sequence of positive real numbers so that ∑_{n=1}^{∞} a_{n} = ∞.

(a) Prove that

∑_{n=1}^{∞} \frac{a_{n}}{1 + a_{n}} = ∞.

(b) Prove that

∑_{n=1}^{∞} \frac{a_{n}}{1 + n^{2}a_{n}} < ∞.

(4) Define

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

wherever the series converges.

(a) Determine the domain for f.

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

(5) Define T : R^{2} → R^{2} by

T (u, v) = (uv, \frac{u^{2} − v^{2}}{2} ).

(a) Compute the Jacobian of T.

(b) Note that T (0, 1) = (0, −\frac{1}{2}). Do there exist neighborhoods A of (0, 1) and B of (0, −\frac{1}{2}) such that there is a differentiable function S : B → A satisfying S(T (a)) = a for all a ∈ A? Briefly explain your reasoning.

(c) Is T one–to–one?

(d) Sketch R = T ([0, 1] × [0, 1]).

(e) Compute the integral

∫_{R} y dxdy.