Math 104 homework 5 consider two-dimensional space r2 with

Question

Math 104: Homework 5-

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

_n=2∑^∞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.

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

d₁(x, y) = (x - y)²,  d₂(x, y) = √(|x - y|),  d₃(x, y) = |x² - y²|,  d₄(x, y) = |x - 2y|.

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

3. Consider two-dimensional space R², with positions written as u = (u₁, u₂), and the Euclidean norm defined as ||u|| = (u₁² + u₂²)^1/2. The Poincare disk model consists of the points X = {u: ||u|| < 1}, with metric

d(u, v) = cosh^-1[1+(2||u - v||²/(1 - ||u||²)(1 - ||v||²))]

for all u, v ∈ X. Define r = cosh^-1 5/4. Draw the Poincare 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.]

4. Suppose that (p_n) is a Cauchy sequence in a space X with metric d, and that some subsequence (p_nk) converges to a point p ∈ X. Prove that the full sequence (p_n) converges to p.

5. 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 (an) 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.