Homework 4. Due on October 16

1. Let x_{1} > 1, and x_{n+1} := 2− \frac{1}{x_{n}} for n ∈ ℕ, Show that \{x_{n}\} is bounded and monotone. Find the limit.

2. Let x_{n} := n^{\frac{1}{n}} for n ∈ ℕ.

(a) Show that x_{n+1} < x_{n} if and only if (1 + \frac{1}{n})^{n} < n, and infer that the inequality is valid for n sufficient large. Conclude that {x_{n}} is ultimately decreasing and the limit a := limx_{n} exists.

(b) use the fact that the subsequence x_{2n} also converges to prove that a = 1.

(c) Find the limit of \{(3n)^{\frac{1}{2n}} \}.

Hint: You are allowed to use the fact that e = lim(1 + \frac{1}{n})^{n}.

3. Show directly from the definition if each of the following is or is not a Cauchy sequence:

(a) x_{n} = \frac{n+1}{n}, (b) x_{n} = 1 + \frac{1}{2} + .... + \frac{1}{n}, (c) x_{n} = n + \frac{(−1)^{n}}{n}.

4. Let \{x_{n}\} be defined by x_{1} := 1, x_{2} := 2 and x_{n} := \frac{1}{2}(x_{n−2} + x_{n−1}) for n > 2. Show that 1 ≤ x_{n} ≤ 2 for all n and that it is a Cauchy sequence. What is the limit of the sequence?

5. Prove that the convergence of ∑ a_{n} implies the convergence of ∑ ^{\frac{\sqrt{a}_{n}}{n}} if a_{n} ≥ 0.

6. Problem 6 on page 78 of Rudin.

7. Problem 8 on page 79 of Rudin.

8*. Problem 16 on page 81 of Rudin.