Math 4032-1, Spring 2024: Information for Students

Spring 2024

General Information for Math 4032--Section 1.

Time 10:30 -- 11:20 AM, M W F. Our class meets from Wednesday January 17, 2024, through Friday, May 3, 2024. We will meet in Room 138 Lockett. Our final exam will be Fri., May 10, 12:30pm - 2:30pm.
Location Room 138 LOCKETT

Leonard Richardson

Office 386 Lockett

Office Hours
  1. MWF 11:30 AM -- 2:30 PM in person in my office, 386 Lockett Hall. Please use these in-person office hours provided you feel healthy. If you feel ill, please use the following online office hours:
  2. TTh online only, by appointment, at this Zoom link: Just email in advance letting me know what hours this afternoon you are available and I will reply with a time to meet. There is a Zoom Waiting Room in case more than one person comes at the same time. However, I can meet with several of you at the same time if the students are comfortable with this.
I am available at many other times. Email first to make sure I'm able meet with you. I answer email many times daily---usually quickly.
Telephone 578-1568

Richardson, L., Advanced Calculus: An Introduction to Linear Analysis, John Wiley & Sons, 2008. ISBN 978-0-470-23288-0. 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 e-book through the LSU Library at this link. Please download the free e-book as the complete pdf file, so you will have the full text on your computer or on an external drive. Last year some students depended on using the book online but couldn't do their homework because sometimes the library's link to the text was not working. So download the full text as one pdf file. The LSU bookstore sells the hard copy, in case you would like the actual book for your personal library, as do I. But don't let them sell you an e-book since this is available to you free of charge using the link given here.

Graduate Assistant

You will turn in homework in class on the due-date. If you must be absent on the due-date, you may email the scan files of your homework to the grader directly by class time: Mr. Yongho Lee at , who will grade those homework problems that are to be turned in---the ones that are assigned in red boldface in the table below.
The grader will be available to answer questions about the homework grading in 121A Prescott Hall on Friday: 11:30 AM - 12:30 PM.
Please be sure to write your solutions neatly and carefully so that they can be read.

Please Note: If you are turning an assignment in by email, the best way to submit an assignment by email is with a device such as a tablet or a drawing board that enables you to write on the computer screen and save or convert to pdf. If you have no such device, you can use a scanner or a phone to photograph your work as jpg images. Then place the images, photographed in the correct order, on your computer screen. Highlight the whole group of pages and select print and then sselect print or save to pdf. That should make one pdf file with all your pages in order. Thank you.

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.


Both Mathematics 4031 and either 2085 or 2090, or equivalents.


Attendance is required and students will be responsible for the classwork on examinations. Each student's presence and participation in class is an essential part of this course. Do not miss class without a valid excuse. When you are absent, you are missed. If you must miss class, please keep track of where we are in the syllabus online, and be sure to visit my daily office hours so that I can help you to keep up with the work you missed.
Statistics show a strongly postive correlation between regular attendance and test grades. 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 encourage that every student comes to class even when the going gets tough.
I will require documentation always 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 to be turned in for grading, email a pdf file or a clear photographic image 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: approximately 3 assignments every two weeks. The assignments are your main work in this course. 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 10-point scale will be added to your Exam average to produce your final average. For example, 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. 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, then at best 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. Moreover, I have seen some of these illegal and unauthorized solutions to problems in my book for sale online that were utterly wrong and must have been written by someone incompetent in mathematics. Buyer beware!! 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. Be aware that when I am writing on the whiteboard I have no way of seeing your raised hand. So speak up with your questions! Our class size is small so you should view our meetings as a two-way discussion and not a formal lecture. It is a good thing to speak up with your questions!
Ask questions after class. Ask questions in my office hours. Ask questions by email. Please consider this: I have been teaching as a University faculty member for more than half a century. 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. It is a pleasure for me to help each and every student, so please give me a chance to help you to the best of my knowledge and ability. Every single one of you is important to me.

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 the classroom, doing so as quietly as you are able so as not to disturb other students. You should have as much class time as possible, so please just come in--quietly--and join the class even if you are late. Also, if homework is due that day, remember to turn it in on paper to me. Class time is a time for work. So when class begins please turn your attention to the work of the class. Because of the covid-19 pandemic, it is recommended that you choose the same seat each day in the classroom. This is to minimize the number of students who might possibly expose you to infection. Of course, everyone who is exposed or ill should get tested and follow LSU's reporting and quarantining instructions. Thank you.


These will be closed-book tests: No notes, whether on paper or electronic, are allowed. No communication devices are allowed. 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 problems---including those not collected---together with examples from the lectures and notes. Thus if you have been coming to every class and 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.


We plan to have two or 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 hour test grades divided the maximum possible cumulative score, expressed as a percentage. Let HA denote your homework average on a 10 point scale. Your Final Average FA will be FA = TA + HA . (Alternatively, if it will benefit you, instead of adding your HA to your TA, we will replace your lowest test grade with your HA converted to a 100% scale. But experience shows most students benefit most from the calculation with TA + HA.) Thus 0<= FA <= 110. The minimum grade for each letter grade is as follows:
A+, 97
A, 93
A-, 90
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 hint---either in class or in my office---or 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. Non-attendance 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 learn---but 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.
  • Caution about Online Sources: Many of us make use of online information occasionally. But students need to be aware that your homework must be logically consistent with the development of Advanced Calculus in our course. What is an exercise in one course may be a theorem in another course. And the order of logical development varies from one course to another. Also, remember that online sources, such as UTube, are not universities. They have no faculty and no academic accreditation. Also, proofs generated by artificial intelligence are apt to be completely lacking in logic. They are just chains of words that commonly follow one another somewhere in a data base. When it comes to learning advanced calculus, there are no shortcuts. We need to do our own work to cultivate a strong sense of logical reasoning.

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 due-date appropriate to this semester appears in the left-hand 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.

Due Date

Assignments: Hand in problems in red bold face for grading. The problems in red bold face are required. Assignments must be written neatly so that the grader can read them. There is also a class of optional problems, called Bonus Problems, which are intended for those students who find the required homework easy and want to be seriously challenged. These are worth up to 20 extra homework points per problem. Bonus problems need to be emailed in pdf format directly to me, your teacher, on a separate sheet from the regular homework, clearly marked Bonus Problems at the top. 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. Bonus problems must be handed in separately from the normal homework, and they will be graded more strictly for logical rigor than the required homework. Please read the Academic Honesty policy above!

January 19 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.
January 22 We will continue our brief review of integration, including the Darboux Integrability Criterion.
January 24 3.18, 3.19, 3.27. (not to hand in)
January 26 3.26 ( Hint: Take note of how we handled Exercise 3.19 in class Wednesday.)
January 29 3.32 -- 3.39.
January 31 3.40, 3.42, 3.44. Hint for 3.44(a): To show that the infinite set B={xk | k in N} is a basis, show 2 things: every polynomial is a linear combination of finitely many elements of B, and no linear combination of finitely many elements of B can be the zero function unless all the coefficients are zero.
February 2 3.45, 3.46, 3.48--3.51, 3.53 .
February 5 3.47, 3.52, 3.58.
February 7 Finish doing these problems: 3.45, 3.46, 3.48--3.51, 3.53.
February 9 Please do problems 4.1, 4.3, 4.4, 4.6, 4.8.
February 16 Hand in: 4.5, 4.7.
February 19 4.9 ; 4.12--4.14, 4.16, 4.18.
February 21 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(xn) converges. One must prove that the limit is independent of the choice of xn-->a+. Note that in 4.21 you need to justify your true/false answer with a proof of your conclusion! Optional 20-point Bonus Problem B1: 4.19. If you decide to do the Bonus Problem, please email it to me,, as a pdf file one week from this date. Do not submit it to the grader: just email it to me. I grade the bonus problems.
February 23 4.24 --- 4.28;
February 26 Hand in: 4.29, 4.32.
February 28 . Bring questions to review for Hour Test #1.
March 1. First Hour Test today. This test will cover all assigned work that was due before today.
March 2 . Please download Hour Test #1, 2024, Solution Sketches and Class Statistics.
March 4. 4.35--4.37.
March 6 Hand in: 4.38. Also: 4.40, 4.42 -- 4.45. Optional 20-point Bonus Problem B2:4.39. In 4.39, you may assume C1[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 C1[a,b]. You may use the fact that the sup-norm is a norm, from Math 4031. If you decide to do the Bonus Problem email it to me at in pdf format one week from this date. It is not part of your regular, required homework.
March 8 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!
March 18 4.47, 4,48, 4.50.
March 20. 4.49, 4.51.
March 22. 5.1, 5.2(b, c: test for convergence only), 5.3 --5.5.
March 25. 5.6, 5.7. (Since 5.7 has 6 parts, it will count double: 20 points instead of 10. Hint for 5.7c: The Geometric Series formula (5.1) will be helpful.) Also: 5.2
March 27. 5.9, 5.10--5.15, 5.19. . For problem 5.9, modify the instructions as follows: Use either the comparison test or the limit comparison test stated in Exercise 5.8 to determine the convergence or divergence of each series. .
March 29 Good Friday Holiday
April 1. 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.
April 3. Second Hour Test today! This test will cover the work that was due after the first hour test and up to the present.
April 4. Please download Hour Test #2, 2024, Solution Sketches and Class Statistics.
April 8 5.20, 5.24 -- 5.25.
April 10. LSU Campus Closed Today for Weather Emergency! Here is a link to the Zoom lesson for April 10.
April 12 . 5.21, 5.23, 5.29. Also: 5.27. Optional 20-point Bonus Problem B3: Let x_k be a conditionally summable sequence. Let a and b be any two real numbers with a less than b. Let S_n(x) be the nth partial sum of the series x_k, and use S_n(y) in the same way for y_k. Show:
(i) There exists a rearrangement y_k of x_k such that the S_n(y) --> = a.
(ii) There exists a rearrangement z_k of x_k such that limsup S_n(z) =b and liminf S_n(z) =a.
April 17 5.30-- 5.31, 5.36.
April 19 5.37, 5.38. You will need to use Definition 5.4.3 for Problem 38, parts c and d.
April 22 5.40-- 5.43.
April 24. 5.39, 5.44
April 26. Bring questions to review for the third hour test!
April 29. Third Hour Test today. This test will cover work done since the second hour test.
April 30 . Please download Hour Test #3, Spring 2024, Solution Sketches and Class Statistics. Study the solutions!
May 1. 5.46.
May 3. Bring questions to review for the Final Exam! Also, please remember to fill out the end-of-course evaluation form that is available to you online! Your anonymous feedback is very important: It helps your teacher and the University to serve your needs to the best of our ability. Remember: Your opinion matters: You are the reason we are here! Thank you..
Don't forget to review from the beginning
of the course! This 200-point exam will cover the whole course in a uniform manner, so remember to review from the beginning of the course. Your final grade for the course will be the larger of the following two: 1. The grade guaranteed by the formula provided higher on this page. 2. One letter below the final exam grade. Thus the final exam provides a safety net that supplements the calculations specified above. As an extreme example (that has not actually occurred in my experience) one might have a failing final average and an A on the final exam, resulting in a B for the course. But less extreme examples do in fact occur, showing the value of catching up by the end of the course.

Exam Week Exam week Office Hours: MW 1PM -- 3PM, and F 3PM -- 5:00 PM in 386 Lockett Hall. Email me as ususal if you have questions and can't make it to my office hours. I can arrange a Zoom office hour for Tuesday late afternoon by request. Email me as ususal if you have questions and can't make it to my office hours.
Fri., May 10 Our final exam will be Fri., May 10, 12:30pm - 2:30pm.
May Please download Final Exam, Spring 2024, Solution Sketches and Class Statistics.
Possible additional topics .6.1, 6.2, 6.7, 6.9 -- 6.11.
Possible additional topics. 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].)
Possible additional topics. 6.14, 6.20. 6.13, 6.15, 6.18, 6.19, 6.24--6.26.
Possible additional topics . 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.)
Possible additional topics. 6.36, 6.43.; Optional 20-point Bonus Problem: 6.29. (Hand in separately from other homework, as usual.)