Spring 2015 
General Information for Math 40321 
Time  MWF 1:30 PM. 
Calendar  Our class begins on Wednesday, January 14. The Final Exam will be on Friday, May 8, 10:00 AM  NOON 
Location  Room 113 Lockett Hall 
Leonard Richardson 
Office 386 Lockett 
Office Hours  MWF 12:001:00 PM. I am available at many other times. Call first to make sure I'm able meet with you. I answer email many times dailyusually quickly. 
Telephone  5781568 
EMail  rich@math.lsu.edu 
Text  Richardson, L., Advanced Calculus: An Introduction to Linear Analysis, John Wiley & Sons, 2008. ISBN 9780470232880. This text is also an online ebook in the LSU Library. 
Graduate Assistant  Mr. Sijing Liu will grade your homework and hold office hours on Monday and Friday from 3:304:30 PM in 361 Lockett. 
Free Math Tutoring 
Free Math tutoring is available in 141 Middleton library. The hours of operation are: Monday  Thursday, 9:30 AM  7:00 P.M., and Friday, 9:30  3:00 PM. While at times it is crowded you can sit and work your homework and get help when needed. 
Math Major Requirements and Recommendations
Either Math 4032 or Math 4035, preceded by Math 4031, satisfies the Advanced Calculus requirement for the Mathematics major with a mathematics concentration. It provides strong preparation in analysis for graduate study of mathematics and for applications. The Department very strongly recommends that Mathematics majors planning graduate study in Mathematics take all three Advanced Calculus courses: Math 4031, 4032, and 4035.
Prerequisites
Mathematics 2057, 2085, and 4031, or the equivalent.
HomeworkProblems, mainly proofs, will be assigned frequently. These will be collected, corrected, and returned at the next class meeting.
You are encouraged to seek hints to help you get started with these problems! Please turn in every assignment! The key to learning to prove theorems lies in how you study Advanced Calculus. It is very important to understand thoroughly how and why the proofs presented in the book and in class work. Please read the Introduction to your textbook! We will go over every collected homework problem in class, to help you prepare for tests. At the end of the course, you will have the benefit of whichever one of the following two Homework Bonuses gives you the best final average.
 Bonus #1: Add your average homework score on a 10point scale to the average of your 3 hour tests with the final exam. For example, if your average on the homework is 5 points out of 10, and you have an 85% exam average, Bonus #1 raises your final average to 90%, which is an A.
 Bonus #2: Replace your lowest hourtest grade with your adjusted homework average. This will be calculated as follows. Let S denote your homework average on a 10point scale. Your adjusted homework average HW=10*S so as to place your homework average on a 100point scale.
Proofs assigned for homework are a very important learning experience. Some students try an easier technique  copying the correct proofs from the board after the homework has been graded, without turning in their own efforts. This tends to produce low grades on Part I of each test, because the student's own conceptual errors have not been turned in and thus have not been corrected. So please turn in every assignment!TestsThese will be closedbook tests: No books or notes are permitted. Part I of each hour test will consist of a choice of 8 out of 12 short answer questions, and Part II will offer a choice of 2 out of 3 proofs. (The Final Exam will be equivalent to two hour tests.) The proofs will be modeled closely on the collected homework, and they are sometimes identical. The short questions will be small variations of homework problemsincluding those not collectedtogether with examples from the lectures and notes. Thus if you have done the homework conscientiously, you should be prepared well for all tests. If you must miss a test, it is your responsibility to speak to me as soon as possible to determine whether or not your excuse is acceptable.GradesThere will be three hour tests, worth 100 points each, and a two hour final examination, worth 200 points. Your test average will be the sum of all your grades divided 5. Your Final Average will be the better of the two grades calculated by adjusting the average using either Pass #1 or Pass #2 described above under Homework.Final Grading Scale: 90 100 (A), 8089 (B), 7079 (C), 6069 (D), Below 60 (F). You should save all your graded work for future study and in case you think your final grade is in error.
Remarks
Many students need help to learn how to write proofs. If you feel confused, it is important to see me for help as soon as possible. If you don't know how to start a homework problem, ask for a hint  either in class or in my office  or even by email. I can guarantee you it is possible to learn to write sound proofs: Learning begins with your efforts and your persistence.
Homework Assignments and DownloadsWe will update the list of assignments and tests below as the semester progresses. You will know an assignment has been updated if a duedate appears in the lefthand column. 
Academic Honesty 
The University has clear policies requiring academic honesty. If you email me about a pending assignment, I may send a hint to the whole class in answer to your question, not giving your name of course! If on the other hand you get a good idea from another book, or from talking with a friend, academic honesty requires that you acknowledge your sources openly. Above all, never copy directly from another person's written work as though it were your own. Remember that your own good name is irreplaceable. This is a sound principle which will serve you well throughout your life. Moreover, on a practical level, it is very foolish claim as your own an argument from a former student in this class or from a textbook. The arguments which are copied can be recognized very easily as not coming from the student, and often the precise source can be identified readily. This means that the honorable course of action is also the practical one. 
Due Date  Assignments: Hand in problems in red for grading. The problems in red are required. But there is also a class of optional problems, called Bonus Problems. These are worth up to 20 extra homework points, if turned in on a separate sheet from the regular homework. Bonus problems are due, if you choose to do one of them, one full week after the date listed, unlike normal graded homework, which is due the date listed. 
January 16. 
Read Sec. 4.1. Make sure you can solve 4.1, 4.3, 4.4, 4.6, 4.8 
January 19. 
Martin Luther King Holiday. No Classes. 
January 21. 
Hand in: 4.5, 4.7, 4.9. 
January 23.  4.12, 4.16, 4.18. 
January 26. 
4.24  4.28; Hand in: 4.15, 4.20, 4.21. Note that 4.20 is challenging. It is not sufficient in part (b) to prove that f(x_{n}) converges. One must prove that the limit is independent of the choice of x_{n}>a+. Optional 20point Bonus Problem: 4.19. If you decide to do the Bonus Problem turn it in one week from this date and give it to me, separately from the normal homework. 
January 28. 
Hand in: 4.29, 4.32. Also: 4.354.37. 
February 2.  4.40, 4.42  4.45. Hand in: 4.38. Optional 20point Bonus Problem:4.39. In 4.39, you may assume C^{1}[a,b] is a vector space. Just check that the given norm is a really a norm, and explain why each Cauchy sequence in that norm converges to a function that belongs to C^{1}[a,b]. You may use the fact that the supnorm is a norm, from Math 4031. If you decide to do the Bonus Problem turn it in one week from this date and give it to me, separately from the normal homework. 
February 4. 
Hand in: 4.41, 4.46. In 4.41 it may help to start with n=0 and then give an inductive proof for general n. In 4.46, take care to justify each equality by explaining what its validity will depend upon at the end. Also, there is a little trick required, so if at first you don't succeed, try again! 
February 6. 
4.47, 4,48, 4.50 
February 9. 
4.49, 4.51. 
February 11.  5.15.5. 
February 13.  5.6, 5.7. Since 5.7 has 6 parts, it will count double: 20 points instead of 10. 
February 18. 
Bring questions to review for Hour Test #1. 
February 20.  First Hour Test today. This test will cover Sections 4.1  5.1. 
February 22. 
Please download Hour Test #1, 2015, Solution Sketches and Class Statistics. 
February 25.  5.95.15, 5.19. 
February 27. 
5.8, 5.16, 5.18. For 5.16, try to make up a proof along lines that would be analogous to the proof of the Ratio Test. 
March 2. 
5.20, 5.24  5.25, 5.27. 
March 4. 
5.21, 5.23. 
March 6.  Optional 20point Bonus Problem: Let x_k be a conditionally summable sequence. Let a and b be any two real numbers with a less than b. Show:
(i) There exists a rearrangement y_k of x_k such that lim y_k = a.
(ii) There exists a rearrangement z_k of x_k such that limsup z_k =b and liminf z_k =a.
If you do these problems, the due date is March 13. 
March 9. 
5.29 
March 11. 
5.30  5.32. 
March 13. 
5.33, 5.36. 
March 16. 
5.34, 5.37. 
March 18. 
5.40  5.44. Also: 5.39, 5.45. In 5.45, there is a repeated typo. The interval should be [0,1], f should be in R[0,1], and the integral should be from 0 to 1. The constant multiplying the integral should be 2 instead of 1/(pi). Sorry! This correction and others are written more clearly in pdf under Errata at the link to the text book given in the box at the top of this page. 
March 20. 
5.46, 5.49, 5.50. 
March 23. 
5.47, 5.48. 
March 25.  Bring a list of questions to review for the second hour test. By asking questions, you help to make the review session beneficial to the whole class. 
March 27.  Second Hour Test today. This test will cover work done since the first hour test. 
March 30. 
Please download Hour Test #2, 2015, Solution Sketches and Class Statistics. 
April 1. 
5.525.54, 5.56, 5.58 
April 312. 
Spring Vacation. 
April 13. 
5.55, 5.57(a,b), 5.59. Optional 20point Bonus Problems: 5.57(c), 5.60, 5.61, 5.62. These 4 bonus problems may be turned in (all of them or some of them) by April 17, separately from the regular homework, as usual. 
April 15 .  6.1, 6.2, 6.7, 6.9  6.11. 
April 17 . 
6.3, 6.4. (Hint: For 6.4, partition the interval [0,T] into two pieces, each of which can be translated by some integral multiple of T to make two pieces of [a,a+T].) 
April 20. 
6.14, 6.20. 6.13, 6.15, 6.18, 6.19, 6.246.26. 
April 22. 
6.27, 6.28. (Hint for 6.28: If a positive Riemann integrable function has Riemann integral = 0, prove that its square has the same integral. Then use this fact.) 
April 24 . 
Bring questions to review for the third hour test! 
April 29. 
Third Hour Test today. 
April 30 . 
Please download Hour Test #3, spring 2015, Solution Sketches and Class Statistics. 
May 1. 
Bring questions for review on May 1. Don't forget to review from the beginning of the course! Also, bring a number 2 pencil on the last day to fill out the endofcourse evaluation form!! 
May 1  May 8.  I will keep my usual office hours during Exam Week, but on a daily basis. I am available at many other times. Call first to make sure I'm able meet with you. I answer email many times dailyusually quickly. Graded bonus problems can be picked up along with the graded remaining homework problems Monday or later. Final Exam: Friday, May 8, 10:00 AM  NOON Please be on time!
The Final Exam will have a choice of 12 out of 18 short questions for 96 points and 4 out of 6 proofs for 104 points. At least one of the proofs and 3 of the short questions will come from the hour tests. There will be two proofs and 6 short questions from each onethird of the semester. Advice: Be sure to review the first twothirds of the course. Often students who are doing well overlook the need for this review. Safety Net Policy: Each student is guaranteed at least the grade determined by the test and final exam average, together with the homework bonus credit. However, as an incentive to do well on the final exam, no final grade will be worse than one letter below the final exam grade. 
May .  Please download Final Exam, Spring 2015, Solution Sketches and Class Statistics. 
