Reference no: EM131081535
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 (an) is a non-increasing sequence of positive real numbers, and that ∑an converges. By considering the Cauchy criterion, or otherwise, prove that nan → 0 as n → ∞.
(c) By considering part (a), find an example of a non-increasing sequence (an) where ∑an diverges, but nan → 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 Nr(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 (pn) is a Cauchy sequence in a space X with metric d, and that some subsequence (pnk) converges to a point p ∈ X. Prove that the full sequence (pn) converges to p.
5. Suppose that (pn) and (qn) are Cauchy sequences in a space X with metric d. Define (an) = d(pn, qn). Show that the sequence (an) converges. It may be useful to consider the triangle inequality
d(pn, qn) ≤ d(pn, pm) + d(pm, qm) + d(qm, qn), which is true for all n and m.
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