Calculus Two: Sequences and Series

About this course: Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

Created by:  The Ohio State University

• Taught by:  Jim Fowler, PhD, Professor

  Mathematics

Commitment 13 hours of videos and quizzes Language English, Subtitles: Chinese (Simplified) How To Pass Pass all graded assignments to complete the course. User Ratings 4.8 stars Average User Rating 4.8See what learners said Coursework

Each course is like an interactive textbook, featuring pre-recorded videos, quizzes and projects.

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Syllabus


WEEK 1


Sequences



Welcome to the course! My name is Jim Fowler, and I am very glad that you are here. In this first module, we introduce the first topic of study: sequences. Briefly, a sequence is an unending list of numbers; since a sequence "goes on forever," it isn't enough to just list a few terms: instead, we usually give a rule or a recursive formula. There are many interesting questions to ask about sequences. One question is whether our list of numbers is getting close to anything in particular; this is the idea behind the limit of a sequence.


18 videos, 5 practice quizzes expand

1. Video: How Can I Succeed in This Course?
2. Video: What is a Sequence?
3. Video: How is a Sequence Presented?
4. Video: Can the Same Sequence be Presented in Different Ways?
5. Practice Quiz: Practice Quiz: What is a Sequence?
6. Video: How Can We Build New Sequences from Old Sequences?
7. Video: What is an Arithmetic Progression?
8. Practice Quiz: Practice Quiz: What are Some Examples of Sequences?
9. Video: What is an Geometric Progression?
10. Video: What is the Limit of a Sequence?
11. Video: Visually, What is the Limit of a Sequence?
12. Practice Quiz: Practice Quiz: What is the Limit of a Sequence?
13. Video: Is it Easy to Find the Limit of a Sequence?
14. Video: For Some Epsilon, How Large Need N Be?
15. Practice Quiz: Practice Quiz: Why Do We Care?
16. Video: How Do Sequences Help with the Square Root of Two?
17. Video: When is a Sequence Bounded?
18. Video: When is a Sequence Increasing?
19. Video: What is the Monotone Convergence Theorem?
20. Video: How Can the Monotone Convergence Theorem Help?
21. Practice Quiz: Practice Quiz: What Other Properties Might a Sequence Have?
22. Video: Is There a Sequence That Includes Every Integer?
23. Video: Is There a Sequence That Includes Every Real Number?

Graded: Review for Sequences

WEEK 2


Series



In this second module, we introduce the second main topic of study: series. Intuitively, a "series" is what you get when you add up the terms of a sequence, in the order that they are presented. A key example is a "geometric series" like the sum of one-half, one-fourth, one-eighth, one-sixteenth, and so on. We'll be focusing on series for the rest of the course, so if you find things confusing, there is a lot of time to catch up. Let me also warn you that the material may feel rather abstract. If you ever feel lost, let me reassure you by pointing out that the next module will present additional concrete examples.


14 videos, 3 practice quizzes expand

1. Video: What Happens in This Module?
2. Video: What Does ∑ aₙ = L Mean?
3. Video: Why Does ∑ₖ₌₀∞ 1/2ᵏ = 2?
4. Video: What is a Geometric Series?
5. Video: What is the Value of ∑ₖ₌ₙ∞ rᵏ?
6. Practice Quiz: Practice Quiz: What is a Series? What is a Geometric Series?
7. Video: What is the Sum of a Telescoping Series?
8. Video: Does the Series ∑ n/(n+1) Converge or Diverge?
9. Practice Quiz: Practice Quiz: What is a Telescoping Series? How Can I Prove That Some Series Diverge?
10. Video: Does the Series 1 + 1/2 + 1/3 + ⋯ Converge or Diverge?
11. Video: Does ∑ sin² k / 2ᵏ Converge or Diverge?
12. Video: What is the Comparison Test?
13. Video: How Can Grouping Make the Comparison Test Even Better?
14. Video: What is ∑ 1/n² ?
15. Practice Quiz: Practice Quiz: What is the Harmonic Series? What About Complicated Series?
16. Video: In What Sense Does 0.99999⋯ Equal 1?
17. Video: In What Sense is ∑ 9⋅10ⁿ Meaningful?

Graded: Review for Series

WEEK 3


Convergence Tests
In this third module, we study various convergence tests to determine whether or not a series converges: in particular, we will consider the ratio test, the root test, and the integral test.


12 videos, 4 practice quizzes expand

1. Video: What Will Happen in This Module?
2. Video: Does Sum n^5 / 4^n Converge?
3. Video: What Does the Ratio Test Say?
4. Video: Does the Ratio Test Always Work?
5. Practice Quiz: Practice Quiz: What is the Ratio Test?
6. Video: Does Sum n! / n^n Converge?
7. Video: How Does n! Compare to n^n?
8. Practice Quiz: Practice Quiz: What is the Ratio Test Good For?
9. Video: Why Don't I Love the Root Test?
10. Video: How Can Integrating Help Us to Address Convergence?
11. Video: How Else Can I Show the Harmonic Series Diverges?
12. Video: Does Sum 1/n^p Converge?
13. Practice Quiz: Practice Quiz: What is the Root Test? What is the Integral Test?
14. Video: Does Sum 1/(n log n) Converge?
15. Video: How Far Out Can You Build a One Sided Bridge?
16. Practice Quiz: Practice Quiz: What Are p-series? How Large Can the Overhang in a Stack of Blocks Be?

Graded: Review for Convergence Tests

WEEK 4


Alternating Series



In this fourth module, we consider absolute and conditional convergence, alternating series and the alternating series test, as well as the limit comparison test. In short, this module considers convergence for series with some negative and some positive terms. Up until now, we had been considering series with nonnegative terms; it is much easier to determine convergence when the terms are nonnegative so in this module, when we consider series with both negative and positive terms, there will definitely be some new complications. In a certain sense, this module is the end of "Does it converge?" In the final two modules, we consider power series and Taylor series. Those last two topics will move us away from questions of mere convergence, so if you have been eager for new material, stay tuned!


15 videos, 2 practice quizzes expand

1. Video: What is this Module All About?
2. Video: Why Have We Been Assuming the Terms are Positive?
3. Video: Why Do Absolutely Convergent Series Just Plain Converge?
4. Video: Why is Absolute Convergence an Important Concept?
5. Video: What is Conditional Convergence?
6. Video: What is an Alternating Series?
7. Video: What is the Alternating Series Test?
8. Practice Quiz: Practice Quiz: What is Absolute Convergence?
9. Video: How Should I Go About Checking the Convergence of a Series?
10. Video: Why is Monotonicity Important in the Alternating Series Test?
11. Video: Why Are Alternating Series Important?
12. Video: Why Is e Irrational?
13. Video: When Do Two Series Share the Same Fate?
14. Practice Quiz: Practice Quiz: What is an Alternating Series?
15. Video: Why Can People Get Away With Writing sum_n a_n?
16. Video: Why is This All so Vague... or Coarse?
17. Video: What Happens if I rearrange the Terms in a Conditionally Convergent Series?

Graded: Review for Alternating Series

WEEK 5


Power Series



In this fifth module, we study power series. Up until now, we had been considering series one at a time; with power series, we are considering a whole family of series which depend on a parameter x. They are like polynomials, so they are easy to work with. And yet, lots of functions we care about, like e^x, can be represented as power series, so power series bring the relaxed atmosphere of polynomials to the trickier realm of functions like e^x.


14 videos, 5 practice quizzes expand

1. Video: What are Power Series?
2. Practice Quiz: Practice Quiz: Introduction to Power Series
3. Video: For Which Values Does a Power Series Converge?
4. Video: Why Does a Power Series Converge Absolutely?
5. Video: How Complicated Might the Interval of Convergence Be?
6. Practice Quiz: Practice Quiz: Where Does a Power Series Converge?
7. Video: How Do I Find the Radius of Convergence?
8. Video: What if the Radius of Convergence is Infinite?
9. Video: What if the Radius of Convergence is Zero?
10. Practice Quiz: Practice Quiz: What is the Radius of Convergence? What if I'd Like a Power Series in Terms of (x-c)?
11. Video: What is a Power Series Centered Around a?
12. Video: Can I Differentiate a Power Series?
13. Video: Can I Integrate a Power Series?
14. Video: Why Might I believe I Have a Power Series for e^x?
15. Video: What Happens if I Multiply Two Power Series?
16. Practice Quiz: Practice Quiz: Can I Do Calculus With Power Series?
17. Video: What Happens if I Transform 1/(1-x)?
18. Video: What is a Formula for the Fibonacci Numbers?
19. Practice Quiz: Practice Quiz: What is a Formula for the Fibonacci Numbers?

Graded: Review for Power Series

WEEK 6


Taylor Series



In this last module, we introduce Taylor series. Instead of starting with a power series and finding a nice description of the function it represents, we will start with a function, and try to find a power series for it. There is no guarantee of success! But incredibly, many of our favorite functions will have power series representations. Sometimes dreams come true. Like many dreams, much will be left unsaid. I hope this brief introduction to Taylor series whets your appetite to learn more calculus.


12 videos, 2 practice quizzes expand

1. Video: What is This Last Module About?
2. Video: What is Better Than a Linear Approximation?
3. Video: What is the Taylor Series for f Around Zero?
4. Video: What is the Taylor Series for f Centered Around a?
5. Video: What is the Taylor Series for Sin Around Zero?
6. Practice Quiz: Practice Quiz: What Are Taylor Series?
7. Video: What is Taylor's Theorem?
8. Video: Why is the Radius of Convergence of 1/(1+x^2) so Small?
9. Video: How is Taylor's Theorem Like a Souped Up Version of the Mean Value Theorem?
10. Video: Approximately, What is cos x When x is Near Zero?
11. Video: How Do Taylor Series Provide Intuition For Limits?
12. Video: What is a Real Analytic Function?
13. Video: How Are Real Analytic Functions Sometimes like Holograms?
14. Practice Quiz: Practice Quiz: What Can I Do With This, in Practice? What Can I Do With This, in Theory?

Graded: Review for Taylor Series

WEEK 7


Final



The final exam is your opportunity to demonstrate everything you have learned in our time together. You have nearly reached the end of the course. Let me tell you that it has been my honor and my pleasure to be one of your guides through mathematics. I very much enjoyed putting this course together, and I look forward to more. I hope we will meet again.

Graded: On-Demand Final Examination