Math 4031-1, Spring 2020: Information for Students

Spring 2020

General Information for Math 4031--Section 1.


2:30 -- 3:20 PM, M W F. Our class meets from Wednesday January 15, through Friday, May 1, 2020 Our final exam will be Thu., May 7, 12:30pm - 2:30pm.


Room 112 LOCKETT

Leonard Richardson

Office 386 Lockett

Office Hours

MWF 1PM-2PM; T Th 11:30AM-12:30AM. Starting March 30 and for the remainder of the semester, you will find me during office hours at this link:

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 daily---usually quickly.





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.

Graduate Assistant

Beginning March 30, you will email the scan files of your homework to Ms. Jinpu Zhou , who will grade those homework problems that are to be turned in---the ones that are assigned in red boldface in the table below. She will be available to answer questions about the homework grading in personal meeting room via Zoom using the link on Mondays from 11:30 am to 12:30 pm.

Please be sure to write your solutions neatly and carefully so that she 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.


Both Mathematics 2057 and 2085, or the equivalent.

Attendance will be recorded daily but will not be counted in your final grade.

Every student's presence and participation in class is an essential part of this course. 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. Attendance will be taken and recorded daily. If you need to be absent from an examination you must tell me why so that I can determine whether or not such an absence is excused. 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 scan 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. 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 10-point 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. 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 question--PLEASE 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. It is a pleasure for me to help students, 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 in--quietly--and 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. Thank you.


These will be closed-book 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. You may have a scientific calculator: However symbolic calculators or graphing calculators are prohibited. 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 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.


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. 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. HOWEVER, unlike the latter services, there are businesses both locally and online that sell their own instructional services to students for fees. LSU does not authorize or approve any such services and does not recognize their competence to serve your educational interests. Advertising for any such commerical service is prohibited in our classroom. If you see such advertising being circulated in our classroom please report it to me. And buyer beware!

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 turned in directly to me 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 13

Today has been dedicated to the tail that wags the dog. Enjoy!

January 15

Read this syllabus and bring any questions you have to class today.

January 17

Read Pp. xxi--7.

Do problems 1.2, 1.4, 1.5, 1.11. These are not to hand in, but you should write your solutions on paper in order to learn from the work. We will go over some---but not all---of these problems in class according to your requests. Let's be partners in this work: You should ask me about the questions you would like me to solve on the board. You will be responsible for all assigned problems, whether collected or not.

January 22

Do problems 1.1, 1.6, 1.8, 1.10. Finish reading section 1--especially the proof of the triangle inequality, which we will use throughout the course.

January 24

Hand in these problems: 1.3, 1.7.

January 25 SATURDAY

Saturday Makeup Class today due to cancellation on January 13.
Do: 1.12 -- 1.16. (Remember: Only problems in RED BOLDFACE are to be handed in!
We will probably be learning the completeness axiom today!)

January 27

Hand in these problems:
  1. Prove that the vector product (also called the cross product) of vectors in 3 dimensional space, does not satisfy an associative law for multiplication. (Hint: find a simple example of a cross product of 3 vectors that fails to be associative, showing the calculations. It is easiest to do this with the standard unit vectors: i, j, and k.)
  2. Use the result of homework problem 1.3 to prove that the real number 1 is positive. That is, prove that 1>0. (Hint: Use the Axiom of Trichotomy to see that there are only 3 potential cases for the number 1. Show that two of those cases lead to an impossibility, leaving only one possibility.)
  3. Do problem 1.9.
Please be sure to write your solutions neatly, so that the grader can read them!

January 29

1.18, 1.20--1.21, 1.23 -- 1.24, 1.26, 1.27.

January 31

1.19, 1.22.
Read Section 1.3 up to Definition 1.3.4. Do exercises 1.31, 1.33, 1.36--1.39.

February 3

1.28, 1.32. Also: 1,29--1.30, 1.34.

February 5

135, 1.41--1.44, 1.47--1.50, 1.53--1.55.

February 7

1.51, 1.57. (Hint for 1.51: Do not compute with (or even write) lim xn or lim yn until you have proven that these limits exist. Otherwise, you will be operating with undefined terms.) (Hints for 1.57: It may help to introduce the notations sn(x) for the supremum of the nth tail Tn(x) of the x sequence, and similarly for the y sequence and the x+y sequence.) Also: do 1.59--1.61, 1.63. Please note that the Bolzano-Weierstrass theorem, the Nested Intervals theorem, and the Heine-Borel theorem will require your focused attention! Here is an optional bonus problem, 1.56, for those students who are looking for more challenging problems than those in the required homework. Turn this in separately from the other assignments one week from the due date printed to the left of this box, if you choose to do it. Bonus problems will be graded more strictly for logical rigor than the required homework. Remember, bonus problems will be collected separately from normal homework.

February 10

1.62, 1.64; Also: 1.67--1.70. Please note that the Bolzano-Weierstrass theorem, the Nested Intervals theorem, and the Heine-Borel theorem will require your focused attention!

February 12

1.71, 1.72. Here is another optional bonus problem, for one week from the date to the left if you choose to do it. This question is similar to 1.70, but not identical! Suppose an is a strictly increasing sequence, bn is a strictly decreasing sequence, and an < bn for all n. Prove or Give a Counterexample: The intersection from 1 to infinity of (an,bn) is nonempty.

February 14

1.76--1.79; 1.80, 1.82, 1.85, 1.86. Be careful with terminology! If E is contained in the union Ua in AOa of open sets Oa then it is the set {Oa | a is in A} that is an open cover of E. Do not confuse the open cover with its own union. The union of a family of sets is only one set!

February 17

1.83, 1.84.

February 19

Bring questions to review for Hour Test #1! Everyone is expected to have a written list of questions. Your questions will help to make the review more useful for yourself and for the whole class. Last year's first hour test is available below for you to download. Note however that what we cover in the course does vary somewhat from year to year, as does the coverage of the tests.

February 21

First Hour Test today. This test will cover all assignments up to this date.

February 24

Please download and read carefully the Hour Test #1, Spring 2020, Solution Sketches and Class Statistics.

February 28

1.87, 1.89, 1.90, 1.92, 1.94.

March 2

1.88, 1.91. Here are two more Bonus Problems, for those who seek more challenging exercises: 1.93, 1.96. If you choose to do these, turn them in separately from the regular homework a week from today. After any bonus problems are graded and returned to you, please feel free to come to my office to ask for correct solutions.

March 4

2.1--2.2, 2.4--2.5, 2.13. Here is another optional bonus problem, for a week from today if you choose to do it: Problem 2.16.

March 6

2.6, 2.7 , 2.8; (Hints: For 2.6, use the sequential criterion to prove the limit of the function does not exist. Do not use L'Hopital's Rule for 2.7(b)! Instead, draw a unit circle and find useful inequalities by comparing areas of triangles and a circular sector, expressible in terms of x, sin x, and tan x.) Also: 2.3, 2.14, 2.15.

March 9

2.19--2.20, 2.22-2.23, 2.25.

March 11

2.21, 2.24, 2.27. For 2.27, follow the sequence of steps given. This exercise is a theorem discovered by Cauchy. For optional bonus credit: 2.26 and 2.28. These two would be for one week from this date.

March 13

2.29, 2.30. 2.32--2.34, 2.40--2.41, 2.43--2.47.

March 23

Please view A recording of our trial run of a Zoom classroom today. The printed transcript of the spoken audio should appear in a few days, when Zoom gets it from its Artificial Intelligence sub-contractor.

March 30

2.35, 2.37, 2.42. In problem 2.42, remember to prove your conclusions for each of the three stated questions! Starting today and for the rest of the semester, you will log into class using this URL: Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
Today's Link. Please Note: The printed transcript on the right is generated by artificial intelligence. You will see here the limitations imposed by the artificiality of artificial intelligence: Computers have no sense of the meaning or purpose of anything. Yet meaning and purpose is what matters so much to human beings in every sphere of human activity. But I hope the transcripts may be useful, and sometimes rather amusing.

April 1

2.48, 2.50, 2.52-- 2.59. Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 1.

April 3

2.49, 2.51. For 2.49, see Example 2.6.
Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 3.

April 6

Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 6.

April 8

2.64, 2.68 . In these exercises, you may use derivatives and L'Hospital's rule although they do not appear in this text until later.

Be sure to turn in your written problems above to Ms. Jinpu Zhou by email before class time, as we will go over these two problems in class today. If you are photographing your problems with a smart phone and saving as jpeg, try using your computer to PRINT these files TO PDF, an option available on many computers. That way you can save to pdf which makes it possible to grade them with markings you will be able to read when they are returned to you.

2.60 --2.63, 2.65 -- 2.67, 2.69. Also, 2.70, 2.73, 2,76, 2.80, 2.82, 2.84, 2.85, 2.88 if you would like to start reviewing!
Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 8.

April 10

Second Hour Test is available online today! Here is how it will work. Since this day is a University holiday, no new work will be done, but I will conduct a voluntary review class meeting via ZOOM. No new work will be done at the voluntary review. After the voluntary review class I will place the take-home hour test online at the class website in the homework line for today's date, so you can download a copy. Please print a copy of the test, work it out following the rules on an honor system, and return it by email to ME (NOT to the grader Ms. Zhou, because I grade all the tests) by 6 PM Monday! (If you can return it Friday night or at least earlier than Monday, that will help me very much to get the grading finished more rapidly.)
If you have no scanner and need to photograph your work with your phone and save as jpeg, then try this please: put the jpeg files into your computer, highlight the whole group of pictures, right click PRINT and then select PRINT TO PDF. That way I can receive a multipage PDF file which is possible to grade in a way you will be able to read later. Thank you! Please let me know if there are any problems with this test arrangement.

Please download a copy of the Take-Home Hour Test from this link: Take-Home Hour Test #2
Today's Voluntary Review Class may be viewed at this link in the cloud: LINK for April 10 Voluntary Review Class.

April 13

Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 13.
If you forgot to attend class today, be sure to watch this recording since we have started Riemann Integration today and you will need this information for the homework due Wednesday.

Please download Hour Test #2, Spring 2020, Solution Sketches and Class Statistics.

April 15

3.3. Be sure to email the solution to 3.3 to the grader before class time, at 2:30 PM, since I will go over the problem in class today.
Also: 3.2, 3.4--3.9, 3.11, 3.12. I think you are ready to do problem 3.3 to email to the grader Wednesday. The problems that are not to hand in depend on theorems we have not yet discussed in our online classes, but we will discuss those theorems and these exercises Wednesday. Here is an optional bonus problem that would be due 1 week from Wednesday: 3.14. Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 15.

April 17


Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 17.

April 20

Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 20.
April 22

3.25, 3.26, 3.30b. ((Hint for 3.25: In 3.25, the strictness of the inequality is the whole problem.) (Hint for 3.30b: Only part (b) is assigned, but here is a hint anyway for the unassigned part (a): In 3.30(a) one of the directions of implication has already been proven in the text. The other direction remains to be proven. (Hint: Use theorem 3.2.4, the variant of the Darboux Integrability Criterion.) Also: 3.18--3.20, 3.22, 3.23.
Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 22.

April 24

3.21, 3.24. (Hint for 3.21: On each subinterval of the partition, consider 3 possible cases: f(x) nonnegative on that subinterval, or f(x) nonpositive on that subinterval, or f(x) has both positive and negative values on that subinterval.) Also: 3.27.
Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 24.

April 27

3.35. Also, 3.34, 3.36, 3.37, 3.38.
Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 27.

April 29

3.40. In problem 3.40, finding a value of K and showing that it works as claimed is what is meant by showing that T is bounded. Also: 3.39, 3.41, 3.42, 3.44.
Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for April 29.

May 1

Bring questions to review for the Final Exam! 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.
Here is the link to today's class recorded on Zoom and available to you for 30 days from today:
LINK for May 1.

Wed., May 6

Please download your copy of the Final Exam, Spring 2020. Please read the instructions carefully and return your finished exam to me tomorrow as indicated in the instructions. Let me know if there are any difficulties and Good Luck!

May 10

Please download Final Exam, Spring 2020, Solution Sketches and Class Statistics.