Master’s Comprehensive Exam – Advanced Calculus – January 1992

1. (a) Define a sequence p_{n} of points is a Cauchy sequence.

(b) Show that any convergent sequence is a Cauchy sequence.

(c) For this part you are to prove part of the converse of part (b). Assume that p_{n} is a Cauchy sequence and assume that every bounded monotonic sequence is convergent. Construct the point p which is the limit point of p_{n}. (Note: Just describe the method by which we know that this point exists – you do not have to show that it is the limit.)

2. (a) Define a set S is n-space is compact.

(b) Show that any compact set is bounded.

3. (a) Define the double integral ∫ ∫_{R} f exists and has value v where R is a rectangle.

(b) Let f : [a, b] → R be continuous. Show that for any ϵ > 0 there is a δ > 0 such that if N is a partition of [a, b] with mesh d(N) < δ then the difference between the upper and lower Riemann sums satisfies |\overline{S}(N) − \underline{S}(N)| < ϵ

4. (a) Define f_{n}(x) converges uniformly to f(x) on [a, b].

(b) Let f be a continuous function on [0, 1] with f(1) = 0. Show that f(x)x^{n} converges uniformly to 0 for x ∈ [0, 1].

5. Suppose that f′(x) is continuous on the real line R. Show that for any finite interval [a, b] there is a constant K such that if x ∈ [a, b] and y ∈ [a, b] then |f(x) − f(y)| < K|x − y|.

6. Does the following series converge pointwise on the reals R? Does it converge uniformly on R?

∑_{n=1}^{∞} \frac{1}{n^{2} + x^{2}}

7. Consider F (x, y, z) = −x^{2} + 3y^{2} + 2y^{3}z^{2} = 0. If p_{0} = (1, −1, 1) then F (p_{0}) = 0.

(a) Does F (x, y, z) = 0 define a function z = ϕ(x, y) in some neighborhood of (x, y) = (1, −1) which satisfies F (x, y, ϕ(x, y)) = 0? Why?

(b) Does F (x, y, z) = 0 define a function y = ψ(x, z) in some neighborhood of (x, z) = (1, 1) which satisfies F (x, ψ(x, z), z) = 0? Why?