## 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 {pn} converges to p if and only if every subsequence of {pn} 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≤xn≤sn for n≥N, where N is a natural number, and if sn→0, then xn→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 sn denote the sequence of partial sums. Here are three possible approaches:
1. Show that the sequence {sn} is a Cauchy sequence. (This is essentially Rudin's proof.)
2. Use the Nested Interval Theorem (Theorem 2.38).
3. Apply the Monotone Convergence Theorem (Theorem 3.14) to the subsequences {s2n-1} and {s2n}.
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