Math 3100 - Sequences and Series - Spring 2019

An Introduction to Analysis

Neil Lyall
Boyd GSRC 602A
lyall (followed by
25118 Class:
41268 Class:
  Office Hours:


 9:05-9:55 (Boyd 323)
11:15-12:05 (Boyd 323)
 8:30-9:00 & 10:00-11:00

 Lecture notes based on the material covered in class will be produced as we proceed through the semester:

Lecture Notes

Additional References:    1. Sequences and Series, by Malcolm Adams
             pdf (last updated 11/14/16)
  2. Elementary Analysis, by Kenneth A. Ross (Second Edition)
             Free Online Version
  3. Understanding Analysis, by Stephen Abbott (Second Edition)
             Free Online Version

About this course: You will find this course to be very different from the more computationally based courses at the 2000 level (like Calculus). This course is meant to help the transition to the more abstract, theoretical courses at the 4000 level and above. Not only will you be expected to learn this material at the computational level, but you will also be studying the proofs of the theorems and learning to write in a rigorous mathematical style. Because you will be looking at mathematics at a much deeper level than you may have in the past, this course will be very challenging. You must never settle for just ending the right answer to a question, you must make sure that you really understand why that answer is correct, and then, you must strive to communicate that understanding in a clear and concise way.

Homework will be collected once a week.
  1. Homework 1   (due on Wednesday January 16th)
  2. Homework 2   (due on Friday January 25th)
  3. Homework 3   (due on Friday February 1st)
  4. Homework 4   (due on Friday February 8th)
  1. Homework 5   (due on Monday February 25th)
  2. Homework 6   (due on Friday March 1st)
  3. Homework 7   (due on Friday March 8th)
  1. Homework 8   (due on Friday April 5th)
  2. Homework 9   (due on Friday April 12th)
  3. Homework 10 (due on Monday April 22nd)
Quizzes: There will be a short quiz two or three times during the semester, these will be announced ahead of time in class.
No make up quizzes will be given.
Exams: There will be three in-class "midterm" exams and a final exam.
 ** dates are provisional and subject to change **

Exam 1:   Wednesday the 13th of February  Exam 1   Exam 1 Study Guide    Sample Old Exams:  Version 1    Version 2
Exam 2:   Wednesday the 27th of March
Exam 2  Exam 2 Study Guide    Sample Old Exams:  Version 0    Version 1    Version 2
         Exam 3:   Friday the 26th of April Exam 3
 Exam 3 Study Guide    Sample Old Exams:  Version 1    Version 2          
Final Exam:
25118 Class: Wednesday the 8th of May from 8:00-11:00
41268 Class: Mon
day the 6th from 12:00-3:00

        Practice Final Exam

Grading: Homework/Quizzes: 15%                   
Tests: 45% (15% each)
Final: 40%

For full credit, full work must always be shown. Any absence on a test day will result in a test grade of 0. It will be possible to make up for a missed test only if documented justification for the absence is provided.

Attendence policy: The official attendance policy of the university states: 
Students are expected to attend classes regularly. A student who incurs an excessive number of absences may be withdrawn from a class at the discretion of the professor (

In this class, we interpret "excessive" to mean four or more unexcused absences.

Academic Honesty: As a University of Georgia student, you have agreed to abide by the University’s academic honesty policy, “A Culture of Honesty,” and the Student Honor Code. All academic work must meet the standards described in “A Culture of Honesty” found at: Lack of knowledge of the academic honesty policy is not a reasonable explanation for a violation. Questions related to course assignments and the academic honesty policy should be directed to the instructor.

The course syllabus is a general plan for the course; deviations announced to the class by the instructor may be necessary

Tentative Course Schedule (This will be continuously revised/modified as needed!)

Week Commencing on Monday

Wednesday Friday

  Irrationality of root 2 and some properties of the reals  Absolute Value, Inequalities, and Induction
Binomial Theorem
Boundedness and Monotonicity of Sequences  
Convergence of Sequences
Examples  Some Consequences of Convergence and Uniqueness of Limits  
 Convergence implies Bounded
 Order Limit Law and statement of Limits Laws 
Proof of Limit Laws and "Squeeze Theorem"  Special Limits

  Subsequences and Least Upper Bounds     Monotone Convergence and Bolzano-Weierstrass Theorems    More on Subsequences: Limit Inferior and Limit Superior

Review  Exam 1 Cauchy Sequences and More on the Axiom of Completeness

  Introduction to Infinite Series   Comparison Test and Examples    More Convergence Tests 

Absolute and Conditional Convergence Ratio Test and Examples Root Test and Power Series
Continuity Examples and the Sequential Characterization Proof of Sequential Characterizations and More Applications
Spring Break
No Class
 Intermediate Value and Extreme Value Theorems
  Integral Test, Alternating Harmonic Series, and Rearrangements  
  Exam 2
Functional Limits and Differentiation
   More Differentiation and Examples
  Interior Extrema Theorem and (Generalized) Mean Value Theorem  Lagrange's Remainder Estimate for Taylor series  
  Differentiating Power Series and Taylor Series  
Examples  Some proofs
Uniform Convergence More on Uniform Convergence  Examples

  Review Review  Exam 3