Spring 2018 
General Information for Math 4032Section 1. 
Time 
10:30  11:20 AM, M W F. Our class meets from Wednesday January 10, 2018, through Friday, April 27, 2018. Our final exam will be Friday May 4, from 10:00AM  Noon.
. 
Location 
Room 285 LOCKETT 
Leonard Richardson 
Office 386 Lockett 
Office Hours 
MWF Noon  1 PM; TTh 1:302:30 PM. I am available at many other times. Call or email 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. There is a list of errata. If you find an error not on this list, please tell me. The text is also available as a free ebook through the LSU Library at this link. 
Graduate Assistant 
Mr. Jagdeep Singh will grade those homework problems that are to be turned inthe ones that are assigned in red in the table below. He will be available to answer questions in his office, Room TBA Lockett Hall, as follows: TBA. Please be sure to write your solutions neatly and carefully so that he can read them. 
Math Major Requirements and Recommendations
Math 4031, followed by either Math 4032 or Math 4035, satisfies the Advanced Calculus requirement for the Mathematics major with a mathematics concentration. It prepares students for graduate study of mathematics and its applications. The Department strongly recommends that Mathematics majors planning graduate study in Mathematics take all three Advanced Calculus courses: Math 4031, 4032, and 4035.
Prerequisites
Both Mathematics 4031 and 2085, or the equivalent.
Attendance is required and will be counted in your final grade.
Every student's presence and participation in class is an essential part of this course. Most LSU students are conscientious and sensible about coming to class unless there is a serious, excusable reason for not being able to do so. However, there is an unfortunate tendency for some students to become discouraged as the term progresses and to cease regular attendance. This happens despite the fact that a student who is feeling discouraged has an especially great need to be in class and to ask questions. The attendance policy is intended to ensure that every student comes to class even when the going gets tough.
Attendance will be taken and recorded daily and unexcused absences will reduce your final average as follows. If UA is the number of your unexcused absences, UA/4 will be subtracted from your final average. If you need to be absent you must tell me why so that I can determine whether or not such an absence is excused. Depending on circumstances I may require documentation for your absences and documentation will always be required if you are absent from an hour test or from the Final Examination. If you are unavoidably absent on a day when homework is due, email a pdf scan file of your homework solutions directly to the grader before class time.
Homework is required and will be part of your final grade
Problems, mainly proofs, will be assigned frequently. The assignments are your main work in this course. The assignments 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! It is required to 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, your homework average on a 10point scale will be added to your Exam average to produce your final average.
For example, if you have no unexcused absences and if your average on the homework is 5 points out of 10, and you have an 85% exam average, your final average would be 90%. In this example the homework credit would raise your grade from B to A. This is an increase of two grade levels on LSU's +/ grading system. However, in this same example, if you had one unexcused absence, then your final average would be 89.75%, for a B+.
Proofs assigned for homework are a very important learning experience. Some students try a shortcut  copying the correct proofs from the board after the homework has been graded, without turning in their own efforts. This tends to produce proofs on tests that are written by rote from memory, and these tend to be lacking in logic and thus incoherent. It results also in 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. Remember that homework is required!
In order to learn the logical structure of advanced calculus, one needs to follow a given set of definitions and theorems from start to finish. If you wish to use other definitions or theorems from a different book, you must also include a proof that the definition or theorem you have chosen is equivalent to the one we used in the course. This will require that you do much more work than is needed to follow the definitions you have been given in our course.
There are unscrupulous businesses online that will sell you solutions to homework problems. If you were to avail yourself of such a service, you would be cheating yourself out of this part of your education. The result will be an unacceptably low grade and very likely the need to repeat the course and pay tuition a second time for the same course. Your learning of Advanced Calculus will come only from your own work. There are no shortcuts. You need to turn in every assignment on time, come to class daily from the first day of the semester to the last, ask questions about everything you do not understand clearly, and ask questions about any errors indicated on your returned homework assignments.
When should you ask questions?
You should ask questions every time you do not understand something and also every time you are curious about something. Ask questions in class. If I am looking the other way and you have a questionPLEASE call out to me so I can have the opportunity to answer your question! Ask questions after class. Ask questions in my office. Ask questions by email. Please consider this: If you are approximately 20 years old, then I have been teaching this subject for approximately 2 and 1/2 times as long as you have lived thus far. So I ask you to consider that I just may be able to help you with whatever is causing you difficulty if you will permit me to do so. Please give me a chance to help you to the best of my knowledge and ability.
Lateness and Classroom Conduct
Please try to arrive on time for class. But sometimes it may be unavoidable to be late. If you are late, please come right into class, doing so as quietly as you are able so as not to disturb other students. I do not want you to wait outside in the hall. You should have as much classroom time as possible, so please just come inquietlyand take a seat even if you are late. If you arrive late and have missed the attendance sheet being passed around the room, be sure to sign it after class. Also, if homework is due that day, remember to turn it in after class.
Class time is a time for work. So when class begins please put away all cell phones, smart phones, head phones, wrist watch communicators, tablets, laptops, etc, and turn your attention to the work of the class. You are a human being  not an android  so I need your realtime living participation in class. Thank you.
Tests
These will be closedbook tests: No books or notes are permitted, electronic, paper, or on any other medium. No internet connected or other communication or electronic devices are permitted other than a watch to check the time. 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.
Grades
There will be three hour tests, worth 100 points each, and a two hour final examination, worth 200 points. Your test average TA will be the sum of your final exam grade and your three hour test grades divided 5. Let HA denote your homework average on a 10 point scale and UA the number of unexcused absences.
Your Final Average FA will be FA = TA + HA  UA/4. Thus 10< FA <= 110. The minimum grade for each letter grade is as follows:
A+, 97
A, 93
A, 90
B+,87
B, 83
B, 80
C+, 77
C, 73
C, 70
D+, 67
D, 63
D, 60
F, below 60
You should save all your graded work for future study and in case you think your final grade is in error.
Please take note:
Number of Dropped Grades: 0
Curve: None.
General Advice
 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 hinteither in class or in my officeor even by email. If you ask me a question about the homework, or if you email such a question to me, I may be able to think of a good hint and then I would email it to the whole class as a hint. I can guarantee you it is possible to learn to write sound proofs: Learning begins with your efforts and your persistence.
 Attend class and ask questions. Nonattendance or lax attendance is usually the first sign of impending academic difficulty. Sometimes a student feels frustrated because of not understanding the classwork. If that is the case, it is necessary to ask more questions. Advanced calculus is a subject you can learnbut you must participate in this work.
 Assignments to be turned in are collected at the beginning of class. If you arrive late, be sure to turn in your homework at the end of class and sign the attendance sheet. Do not turn it in later than that, because it is not fair to the graduate teaching assistant, who will be busy enough with the work of grading the assignments that are turned in at the proper time.
 LSU offers extensive academic support services to help students adjust to the demands of university studies: List of Frequently Used Services.
Homework Assignments and Downloads
We will update the list of assignments and tests below as the semester progresses. You will know an assignment has been updated if a duedate appropriate to this semester appears in the lefthand column. However, sometimes we will assign a problem for a certain date and then postpone it because we don't cover as much as planned in class. So check regularly for updates as to what is due and when. 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! 
Academic Honesty
The University has clear policies requiring academic honesty. If you get an idea from another book or an online source, 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.

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 10. 
Please read this syllabus and ask any questions you may have about it. Also, please be sure you have a copy of the text so you can do the assignments to follow. We will begin with a brief review of integration and we will pick up where we left off in the fall in section 3.2. 
January 12. 
3.18, 3.19, 3.20, 3.22, 3.23. (not to hand in) 
January 15. 
Martin Luther King Holiday. No Classes. 
January 22. 
3.21, 3.24, 3.26 (to be handed in for grading). Also: 3.27 (not to hand in). 
January 24. 
3.32  3.35. 
January . 

January . 

January . 
Read Sec. 4.1. Make sure you can solve 4.1, 4.3, 4.4, 4.6, 4.8 
January . 
Hand in: 4.5, 4.7, 4.9. 
January .  4.12, 4.16, 4.18. 
January . 
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 . 
Hand in: 4.29, 4.32. Also: 4.354.37. 
February .  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 . 
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 . 
4.47, 4,48, 4.50 
February . 
4.49, 4.51. 
February .  5.15.5. 
February .  5.6, 5.7. Since 5.7 has 6 parts, it will count double: 20 points instead of 10. 
February . 
Bring questions to review for Hour Test #1. 
February .  First Hour Test today. This test will cover Sections 4.1  5.1. 
February . 
Please download Hour Test #1, 2015, Solution Sketches and Class Statistics. 
February .  5.95.15, 5.19. 
February . 
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 . 
5.20, 5.24  5.25, 5.27. 
March . 
5.21, 5.23. 
March .  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 . 
5.29 
March . 
5.30  5.32. 
March . 
5.33, 5.36. 
March . 
5.34, 5.37. 
March . 
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 . 
5.46, 5.49, 5.50. 
March . 
5.47, 5.48. 
March .  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 .  Second Hour Test today. This test will cover work done since the first hour test. 
March . 
Please download Hour Test #2, 2015, Solution Sketches and Class Statistics. 
April . 
5.525.54, 5.56, 5.58 
April . 
Spring Vacation. 
April . 
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 .  6.1, 6.2, 6.7, 6.9  6.11. 
April . 
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 . 
6.14, 6.20. 6.13, 6.15, 6.18, 6.19, 6.246.26. 
April . 
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 . 
Bring questions to review for the third hour test! 
April . 
Third Hour Test today. 
April . 
Please download Hour Test #3, spring 2015, Solution Sketches and Class Statistics. 
April . 
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!! 
April 30  May 4.  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 4, from 10:00AM  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. 
