Real Analysis II
MAT 3751
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Welcome to the course pages for Brian Gill's
Winter 2006 Real Analysis II class! Instructor: Dr. Brian Gill Tentative Winter 2006 Office Hours: Course syllabus |
Daily Information and Homework
Past Homework Assignments:
Assignment for Monday, January 9:
- Complete the remaining three tables in the lab handout. For the tables on p. 33 and 38, use the 2nd Maple file that I sent you (the one for products rather than sums). For the table on p. 35, you can use either of the Maple files -- just write down a formula for the reciprocal of the sequence and then enter that as the sequence. You do NOT need to turn these tables in, but please complete them before class on Monday.
Also turn in the following items:
- Prove the Triangle Inequality (stated in class)
- Write a careful, formal proof of Conjecture 1 from the lab.
- State a conjecture about what happens with the limit of the sum of two sequences in the situation where one of the sequences converges and the other diverges. Explain why you think the conjecture is true, but you don't need to write a formal proof (at least not for Monday…)
- Use your tables to help you to write careful formal statements for Conjectures 2, 3, and 4 (immediately following the three tables). For the last table, also turn in answers to the five questions right before and after the table.
You are permitted to turn in a single copy of this assignment for your group or to work on it and turn it in on your own. However, I encourage you to work with your group if it is at all possible.
Assignment for Wednesday, January 11: Turn in the following items from the lab. You MUST turn in a single copy of all of these items for your entire group.
- Turn in careful, detailed responses to all questions in sections 6.3.3 and 6.4.3 of the lab.
- Turn in solutions for questions 1, 3, and 5 from section 6.6 of the
lab.
- For question 5, you should not need to use an "epsilon-N" style proof. It can be proven using the other claims that you have already proven in the lab. Be sure to clearly state when, where, and how you make use of the conjectures from earlier in the lab (which are now theorems since you've proven them) as you write your proof for #5.
Homework for Friday, Jan 20:
- Both the lab write-up for Lab 7 and the assignment for Section 2.3 will be collected on Friday, Jan. 18, so please plan ahead.
- From Lab 7, please work through all of the examples in section 7.2, answer all of the questions in section 7.3, and do questions 1 and 3 from section 7.4.
- Homework from Section 2.3 of Abbot:
Solutions
- Exercises 1, 3, 6, 8, and 9 from section 2.3 of Abbott. (These are listed as 2.3.1, 2.3.3, 2.3.6, 2.3.8, and 2.3.9 in the book)
- You previously (in 3749) did exercise 2.2.1 using epsilon-N proofs. Rework parts (a) and (b) of the exercise without using epsilon-N proofs. Instead, use the Algebraic Limit Theorem along with the theorem about lim(1/n^p) to write the proofs.
Homework for Wednesday, Jan. 25 : Solutions
- Write a proof for the Monotone Convergence Theorem in the case of a decreasing sequence.
- Prove that the series given in class converges and find its sum.
- Turn in exercises 2 and 5 from section 2.4 of Abbott.
Homework for Monday, Jan. 30 : Solutions
- Turn in exercises 1.4.5, 2.5.1, 2.5.3, and 2.5.5.
Homework for Wednesday, Feb. 1: Solutions (2.6.6a only partially done)
- Turn in exercises 2.6.1, 2.6.2, 2.6.4, and 2.6.6a. Read the rest of exercise 2.6.6, but you don't need to turn it in -- just understand the line of reasoning and how it would prove the equivalence of all 5 of these statements.
Homework for Monday, Feb. 6: Partial Solutions
- Turn in exercises 2.4.1, 2.7.1, 2.7.4, 2.7.6, and 2.7.7.
- For exercise 2.7.1, there are three methods for proving the alternating series test, which are listed as part (a), (b), and (c) of the exercise. READ all three parts of the exercise, and then choose ONE method (whichever you prefer) to actually prove and turn in.
Homework for Wednesday, Feb. 8: Solutions
- Turn in exercises 2.7.2a, 2.7.9
Homework for Monday, Feb. 13: Solutions
- Turn in exercises 3.2.2, 3.2.3
Homework for Monday, Feb. 27: Solutions
- Turn in exercises 3.2.1, 3.2.4, 3.2.8, 3.2.9, 3.2.12, and 3.3.5.
Homework for Wednesday, March 1: Partial Solutions
- Turn in exercises 3.3.1, 3.3.2, 3.3.3, 3.3.4, and 3.3.7.
- Extra credit: Turn in 3.3.6. (I strongly encourage you to do this -- it is not that hard, and it is a fascinating exercise)
- When reading Section 3.3, focus on p. 84-85; don't worry about p. 86-87.
Homework for Friday, March 3: Turn in exercises 4.2.1, 4.2.2, and 4.2.3. Solutions
Homework for Wednesday, March 8: Turn in exercises 4.2.9, 4.3.1, 4.3.3, 4.3.7, and 4.5.2. (for 4.2.9, relate it back to the squeeze theorem for sequences, which was exercise 2.3.3 -- you may use the result of that exercise as needed) Solutions
