Dave4Math / Courses / Infinite Sequences and Series

Infinite Sequences and Series

Master infinite sequences and series. In this course, you'll learn about convergence tests, Maclaurin and Taylor series, and differentiating and integrating series.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Have you ever had a teacher so good that you had to run out and tell all your friends? Introducing Dave’s online math course: Infinite Sequences and Series. Dave is recommended by his students because, as they say, he explains everything in great detail on every topic.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in infinite sequences and series.
  • Students who are enrolled in calculus 2 and want to improve their grade.
  • Anyone interested in getting prepared for calculus 3.
  • Anyone wanting to learn about sequences, series, various tests for convergence, power series, and the Taylor and Maclaurin series.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Infinite Sequences and Series

  • Introduction to Sequences
  • Limit of a Sequence
  • Limit Laws for Sequences
  • Squeeze Theorem for Sequences
  • Bounded Monotonic Sequences
  • Infinite Series
  • Convergence of Infinite Series
  • Geometric Series
  • The Harmonic Series
  • The Divergence Test
  • Properties of Convergent Series
  • Integral Test
  • The p-Series
  • The Comparison Test
  • The Limit Comparison Test
  • The Alternating Series Test
  • Absolute Convergence
  • The Ratio Test
  • The Root Test
  • Approx the Sum of an Alternating Series
  • Summary of Tests for Convergence
  • Rearrangement of Series
  • Introduction to Power Series
  • Interval of Convergence
  • Differentiation and Integration of Power Series
  • The Taylor and Maclaurin Series
  • Techniques for Finding Taylor Series
  • The Maclaurin Polynomial
  • Taylor’s Formula with Remainder
  • Representing a Function by a Series

Course Description

We start this interesting course off with a review of sequences. We discuss what sequences are, limits of sequences, limit laws for sequences, and the squeeze theorem for sequences. Next, we consider bounded monotonic sequences and then prove the monotone convergence theorem for sequences.

After this introduction to sequences, we begin a thorough investigation of infinite series. We discuss partial sums and convergence at length as well as geometric series and the harmonic series. Properties of convergent series and the divergence test are also detailed.

We then explore several convergence tests in detail, including the integral test, alternating series test, comparison tests, and the ratio and root tests. For each convergence test, we motivate why it is essential, how it works, and consider several examples.

After that, one of the main topics of calculus is explored: power series. We go into a great deal of attention on power series, including the Talyor and Maclaurin series. In the end, we examine approximation by Taylor polynomials, which is a generalization of linear approximation as studied in calculus 1.

Recommended Prerequisites for Infinite Sequences and Series

I recommend the prerequisite course Vectors and the Geometry of Space. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 2 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.