Homework 5

Sequences and Subsequences

1. Find the limit of the recursive sequence defined by $a_n=1+\frac{1}{a_{n-1}}$ and $a_1=1$ (it's okay to answer this question numerically; no proof is required). Also find the limit of the sequence $a_n=\sqrt{1+a_{n-1}}$ and $a_1=1$. Show that the limit is the same as a solution of the equations $x=1+\frac{1}{x}$ and $x=\sqrt{1+x}$.

2. Does the sequence $a_n=\cos(\pi\cdot(2n+1))$ converge? What about the sequence $b_n=cos(n)$? Provide evidence and/or proof of your claim. Make a conjecture about when the sequence $a_n=\cos(a\cdot n+b)$ converges.

3. Give an example of a sequence containing subsequences that converge to $\pi$, $\sqrt{2}$, and $-1$.

4. Give an example to demonstrate that it is possible for neither the sequence $(a_n)$ nor the sequence $(b_n)$ to converge, but for the sequence $(c_n)$ with $c_n=a_n+b_n$ to converge.

Sequences and Subsequences (Proofs)

5. Formally prove that the sequence $a_n=\frac{1}{n}$ converges to 0. (Meaning: start off with "Let $\epsilon>0$…")

6. Complete problem 26 in the text (Chapter 2). Use a formal proof.

Metric spaces

7. Let M be the set of places in the (48 connected) United States, and define $d(m_1,m_2)$ to be the number of miles required to drive from place $m_1$ to place $m_2$. For example, according to Google maps, the trip from West Point to San Francisco is 2,924 miles. Explain what each of the three properties of a metric mean in this context (e.g. $d(x,y)\geq 0$ means that the distance from one point to another is at least 0 miles).

8. Complete problem 82 in the text (Chapter 2). See page 72 for the definition of "sum metric".

Optional Questions

O1. Let $a_n$ be defined as the ratio of primes to non-primes in the first $n$ integers. For example, $a_5=\frac{3}{2}$ since the numbers 2, 3, and 5 are prime, whereas 1 and 4 are not (1 is usually not considered to be prime). Compute $a_{100}$ and $a_{1000}$ (you can look up a table of prime numbers).