## Math 521

Lecture: 2

Instructor: Ed Hanson

#### Course Policies and Information

A copy of the syllabus with useful information can be found here. Note that this syllabus was last revised on January 29, 2018.

#### Homework

- Homework 1 due Wednesday, February 7: Rudin Chapter 1, exercises 1, 4, 5, 6, 7. Go here for solutions.
- Homework 2 due Wednesday, February 14: Rudin Chapter 1, exercises 8, 9, 11, 12, 18. Go here for solutions.
- Homework 3 due Wednesday, February 21: Rudin Chapter 2, exercises 2, 3, 4, 11. Go here for solutions.
- Homework 4 due Friday, March 2: Rudin Chapter 2, exercises 5, 6, 7, 8, 22. Go here for solutions.
- Homework 5 due Monday, March 19: Rudin Chapter 2, exercises 12, 13. Rudin Chapter 3, exercises 1, 2 (be sure to prove the limit statement that you compute). Go here for solutions.
- Homework 6 due Wednesday, April 4: Rudin Chapter 2, exercise 15. Rudin Chapter 3, exercises 5, 16, 20, 23. Also, prove that the sequence {p
_{n}} converges to p if and only if every subsequence of {p_{n}} converges to p. Go here for solutions.
- Homework 7 due Wednesday, April 11: Rudin Chapter 3, exercises 6, 7, 8. Also, prove that if 0≤x
_{n}≤s_{n} for n≥N, where N is a natural number, and if s_{n}→0, then x_{n}→0. Go here for solutions.
- Homework 8 due Friday, April 27: Rudin Chapter 3, exercise 9. Rudin Chapter 4, exercises 2, 3, 8, 9, 14. Also, prove Theorem 3.43 from Rudin (the so-called Alternating Series Test) using one of the methods outlined below. Rudin's proof relies on a bit of machinery called
*summation by parts*, but there are more direct methods that give more intuition into why this result is true. Let s_{n} denote the sequence of partial sums. Here are three possible approaches:
- Show that the sequence {s
_{n}} is a Cauchy sequence. (This is essentially Rudin's proof.)
- Use the Nested Interval Theorem (Theorem 2.38).
- Apply the Monotone Convergence Theorem (Theorem 3.14) to the subsequences {s
_{2n-1}} and {s_{2n}}.

Go here for solutions.
- Homework 9 due Friday, May 4: Rudin Chapter 5, exercises 2, 17. Rudin Chapter 6, exercises 2, 4. Go here for solutions (updated May 8).

#### Extra problems

- Here are some practice problems to help prepare for the quiz. Solutions are here.
- Here are some practice problems to help prepare for the first exam. Solutions are here.
- Here are some practice problems to help prepare for the second exam. Solutions are here.
- Here are some practice problems to help prepare for the final exam. Solutions are here.

#### Solutions

- Here are solutions for the quiz.
- Here are solutions for the first exam.
- Here are solutions for the second exam, except for problem 3, which is discussed here

**Last updated:** May 8, 2018