Math 104: Homework 5 (due March 4)

1. Ross exercise 14.12

2. (a) By using the integral test, or otherwise, prove that

∑_{n=2}^{∞} \frac{1}{n(log n)^{p}}

converges if and only if p > 1.

(b) Suppose (a_{n}) is a non-increasing sequence of positive real numbers, and that ∑ a_{n} converges. By considering the Cauchy criterion, or otherwise, prove that na_{n} → 0 as n → ∞.

(c) By considering part (a), find an example of a non-increasing sequence (a_{n}) where ∑ a_{n} diverges, but na_{n} → 0 as n → ∞, to show that the converse of part (b) is not true.

3. Consider the following functions defined for x, y ∈ :

d_{1}(x, y) = (x − y)^{2}, d_{2}(x, y) = \sqrt{|x − y|} , d_{3}(x, y) = |x^{2} − y^{2}|, d_{4}(x, y) = |x − 2y|.

For each function, determine whether it is a metric or not.

4. Ross exercise 13.3

5. Consider two-dimensional space ^{2}, with positions written as u = (u_{1}, u_{2}), and the Euclidean norm defined as ||u|| = (u_{1}^{2} + u_{2}^{2})^{1/2}. The Poincaré disk model consists of the points X = {u : ||u|| < 1}, with metric

d(u, v) = cosh^{−1} [1 + \frac{2||u − v||^{2}}{(1 − ||u||^{2})(1 − ||v||^{2})} ]

for all u, v ∈ X. Define r = cosh^{−1} ^{5}/4. Draw the Poincaré disk, and then calculate and draw the neighborhoods N_{r}(u) for u = (0, 0), (^{1}/2, 0), and (^{3}/4, 0). [This can be done analytically, although if you prefer, you can also make use of computer programs to do it – any method of drawing the picture is acceptable.]

6. Suppose that (p_{n}) is a Cauchy sequence in a space X with metric d, and that some subsequence (p_{n_{k}}) converges to a point p ∈ X. Prove that the full sequence (p_{n}) converges to p.

7. Suppose that (p_{n}) and (q_{n}) are Cauchy sequences in a space X with metric d. Define (a_{n}) = d(p_{n}, q_{n}). Show that the sequence (a_{n}) converges. It may be useful to consider the triangle inequality

d(p_{n}, q_{n}) ≤ d(p_{n}, p_{m}) + d(p_{m}, q_{m}) + d(q_{m}, q_{n})

which is true for all n and m.