Paul Pollack

• Final exam learning objectives/review sheet PDF
• Exam #3 learning objectives/review sheet PDF
• Assignment #7 due 11/20PDF
• Assignment #6 due 11/4PDF
• Exam #2 learning objectives/review sheet PDF
• Assignment #5 due 10/16PDF
• Assignment #4 due 10/2 10/4PDF
• Exam #1 learning objectives/review sheet PDF
• Assignment #3due 9/18PDF
• Assignment #2due 9/6PDF
• Assignment #1due 8/26PDF

• 11/22 Exam #3.
• 11/20 Motivation and definiton of Taylor series. An example of a function with a Taylor series not represented by this series at any nonzero point. Examples of Taylor series for $e^x$ and $\sin x$.
• 11/18 Closed forms of power series. Geometric series. Manipulating closed forms with differentiation and integration.
• 11/15 Simple derivatives (constants, powers of $x$). Rolle's theorem and the mean value theorem.
• 11/13 Quotient rule, chain rule, and the extreme value theorem for derivatives.
• 11/11 Introduction to derivatives.
• 11/8 Illustrating the definition of $\lim_{x\to a} f(x)=L$ with examples. Limit laws.
• 11/6 Proof of the Max Value Theorem. Definition of $\lim_{x\to a} f(x)=L$.
• 11/4 Proof of intermediate value theorem. Statement of Maximum Value Theorem.
• 10/30 More "new from old" rules for continuous functions. Polynomials and rational functions are continuous on their domains. Statement of Intermediate Value Theorem.
• 10/28 Continuity is equivalent to sequential continuity. Establishing functions are continuous straight from the definition. The constant multiple rule for continuity.
• 10/25 Proof of the theorem on the general form of the DOC. Start of continuity. For the rest of the course, we will be using Kenneth Ross's book Elementary Analysis as our reference. The link should allow you to download the PDF for free (thanks UGA!).
• 10/23 Power series. Computing domains of convergence. The general form of the DOC (statement and preparation for proof).
• 10/21 Rearranging the alternating harmonic series to sum to zero.
• 10/18 Exam #2.
• 10/16 Proof of ratio test. A mathematical magic trick. Statement of Riemann's rearrangement theorem.
• 10/14 Absolute vs conditional convergence. Statement of ratio test and examples.
• 10/11 Absolute convergence. Absolute convergence implies convergence. Alternating series test.
• 10/9 Proof of integral comparison test. Example: $\sum_{k=1}^{\infty} k e^{-k^2}$ converges.
• 10/7 Eventual Comparison Test. Limit Comparison Test. Statement of Integral Comparison Test and application to $p$-series test.
• 10/4 Completion of discussion of geometric series. Constant multiple and sum rules for series. Basic comparison test.
• 10/2 Convergence of $\sum 1/k^2$. Divergence of $\sum 1/k$ (harmonic series). The $k$th term test for divergence. Start of discussion of geometric series.
• 9/30 Proof of the LUB property of $\mathbb{R}$. Introduction to series.
• 9/23 Cauchy sequences converge. A sequence of real numbers converges to a real number if and only if it is a Cauchy sequence. Introduction to Least Upper Bounds.
• 9/20 Exam!
• 9/18 Peak indices. The Bolzano--Weierstrass theorem: Every bounded sequence has a convergent subsequence. Start of discussion of Cauchy sequences.
• 9/16 Wrap up of limit laws. Start of discussion of "what is reality." The completness axiom: every bounded, monotone sequence of real numbers converges to a real number limit. $\sqrt{2} \in \mathbb{R}$. Statement of intermediate value theorem.
• 9/13 Completion of quotient test. Diverging to $\infty$ and $-\infty$. A ratio test for sequences.
• 9/11 Limit laws: Sum rule, difference rule, product rule. Start of discussion of quotient rule. Review of (bounded)*(going to zero) goes to zero.
• 9/6 Convergent sequences are bounded. Every subsequence of a convergent sequence converges to the original limit. Start of discussion of geometric sequences.
• 9/4 Two more explicit limit examples.
• 8/30 Interlude: triangle inequality. Applications to estimation. A first ``explicit limit'' example.
• 8/28 Eventually increasing sequences. The definition of convergence. Constant sequences have the expected limit.
• 8/26 Eventually increasing sequences. The definition of convergence. Constant sequences have the expected limit.
• 8/23 $\{a_n\}$ is bounded if and only if there is a real number $M\ge 0$ with $-M\le a_n\le M$ for all $n$. Informal and formal definitions of a subsequence. A subsequence of an increasing sequence is increasing.
• 8/21 Sequences as functions. The notions of increasing, decreasing, and monotone. If $\{a_n\}$ is increasing, and $n$ and $m$ are natural numbers with $n\le m$, then $a_n \le a_m$. Bounded above, bounded below, and bounded.
• 8/19 PMI with a different base case. The Principle of Complete Mathematical Induction.
• 8/16 Review of the Principle of Mathematical Induction (PMI).
• 8/14 Go over syllabus [static copy here]. Introduction to sequences.

UGA's CAPA system contains the following Course Description of MATH 3100:

Precise description of the real number system; rigorous treatment of limits and convergence for sequences, series, and functions; continuity and the maximum and intermediate value theorems; differentiation and the mean value theorem; Taylor approximation.

Those are indeed the topics we'll aim to cover this term. But if you really want to understand what sort of fine mess/math you've gotten yourself into, it will be useful to take a step back and think more philosophically. What is this course doing in the major?

There are two answers to this, equally important. On the one hand, this course aims to introduce you to one of primary strands of mathematical thinking, "mathematical analysis." (Many would say that "algebra" and "analysis" are the two cornerstones of mathematics.) Analysis is the that branch of mathematics that grew out of calculus. So why not just call it calculus? We use a different word because we have a different focus. Calculus courses aim primarily to equip you to solve a wide range of problems. Analysis, at its most basic level, is about understanding why those solution methods make sense. That is, we want to understand the theory behind the practice.

A second raison d'être of this course is that you continue the mathematical apprenticeship which you begun in MATH 3200. This includes the construction of carefully reasoned mathematical proofs. You should not expect this to be easy; indeed, as you mature as a mathematician, you will find yourself confused a good deal of the time! (By the time you get to be a professional mathematician, you are constantly confused.) I will do my best to guide you through the difficulties and to help you come out the other side. Of course, this depends a great deal on your own engagement with the material --- both in class and in office hours.

Our primary resource will be the MATH 3100 course notes developed by UGA Professor Emeritus Malcolm Adams. You may download the notes here. These notes were written with this course in mind and we will follow them closely for about 75% of the semester.

This is a difficult course! There is no shame (far from it!) in seeking help when you get stuck. I strongly recommend that you show up at office hours (times TBD); even better, show up and bring a friend! Note that these are intended as collaborative problem solving sessions. As such: You should expect me to give hints and to follow up on ideas you tried, not to simply telegraph answers.

Both at office hours and in class, you should expect that your input will be treated respectfully, by myself and by your classmates. Turning it around, you are expected to show respect and understanding for your classmates’ ideas. Kindness is important --- everywhere and always!

There are three in-class midterm exams as well as a final exam.

• Midterm #1: Friday, September 13 September 20
• Midterm #2: Friday, October 11 October 18
• Midterm #3: Monday, November 18 Friday, November 22
• Final exam: Fri, December 6, 12:00-3 PM (usual classroom)

No make-up exams will be given. The final exam is cumulative.

Your grade is made up of the following weighted components:

• Each midterm: 15% (total of 45%)
• Homework: 25%
• Final exam: 30%

You are expected to participate in class. In particular, attendance in this course is required. More than four unexcused absences may result in you being withdrawn from the course. Keep me posted whenever you have a conflict that requires you to miss class and this should not be an issue.

All exams are in-class, closed book, and closed notes.

Homework will be collected in class, about once each week. Late homework will not be accepted. (If you have a need to turn in HW early, that can be arranged.) Your lowest HW score will be dropped at the end of the term.

On homework, collaboration is allowed and in fact is very much encouraged. Mathematics wouldn't be nearly as much fun if we couldn't talk about it with other people! However, copying (from a textbook or another student), web searches, and AI tools (such as ChatGPT) are not allowed, and you must write your own final solutions independently. Keep in mind that by entering UGA, you have already agreed to abide by the UGA Honor code described in detail at https://honesty.uga.edu/Academic-Honesty-Policy/.

In practice, what this means that you may discuss homework problems and their solutions with your classmates, but you may not turn in a solution unless you understand it yourself. A reasonable rule of thumb is that you should be able to explain your solutions verbally to me (in all their gory detail) if requested to do so.

Students with disabilities who may require special accommodations should talk to me as soon as possible. Appropriate documentation concerning disabilities may be required. If you plan to request accommodations for a disability, please register with the Disability Resource Center. They can be reached by visiting Clark Howell Hall, calling 706-542-8719 (voice) or 706-542-877 (TTY), or by visiting http://drc.uga.edu.

If you or someone you know needs assistance, you are encouraged to contact Student Care and Outreach in the Division of Student Affairs at 706-542-8479 or visit https://sco.uga.edu. They will help you navigate any difficult circumstances you may be facing by connecting you with the appropriate resources or services. UGA has several resources for a student seeking mental health services (https://caps.uga.edu/well-being-prevention-programs-mental-health/) or crisis support (https://healthcenter.uga.edu/emergencies/).

The Federal Family Educational Rights and Privacy Act (FERPA) grants students certain information privacy rights. See the registrar’s explanation at reg.uga.edu/general-information/ferpa/. FERPA allows disclosure of directory information (name, address, telephone, email, major, activities, degrees, awards, prior schools), unless requested in a written letter to the registrar.