About This Chapter
Series - Chapter Summary
We've taken the stress out of studying series in calculus with this easy-to-follow chapter. The video lessons cover calculating arithmetic series, the formula for power series and the definition of p-series. Take your time as you work through this chapter, going back as needed as many times to ensure comprehension. If you need assistance from an expert, feel free to reach out via the Dashboard. After you complete this chapter, you should be able to:
- Calculate arithmetic and geometric series
- Complete practice problems with geometric and arithmetic series
- Give examples of power series and p-series
- Outline the formula for harmonic series in math
- Define Taylor series and the formula used in them
- Describe the steps used in Taylor series for ln(1+x)
- Provide the definition and formula for Maclaurin series
- Explain how to calculate the Maclaurin series for ln(1+x)
- Apply the ratio test for convergence and divergence
- Compare series to test for divergence and convergence
1. How to Calculate an Arithmetic Series
In this lesson, you'll learn how to sum up any arithmetic sequence without having to add up each and every term. You'll never guess the circumstances that led to the discovery of this marvelous formula.
2. How to Calculate a Geometric Series
If you're not satisfied with knowing how many views your YouTube video will get any one day and need to know how many total views you're getting, geometric series are for you! Learn more in this lesson.
3. Arithmetic and Geometric Series: Practice Problems
There are many different types of problems concerning series and sequences - including some that are pretty abstract. From bouncing balls to adding up odd numbers, test your skills here with additional arithmetic and geometric series practice!
4. Power Series: Formula & Examples
A power series gives us what we call an infinite polynomial on our variable x and can be used to define a wide variety of functions. Learn the definition of power series as well as several examples of functions that can be described by them.
5. P-Series: Definition & Examples
This lesson is designed to help you understand a specific type of series called a p-series. You will determine if a series is a p-series, and you will learn to decide if a p-series converges or diverges.
6. Harmonic Series in Math: Definition & Formula
The harmonic series provides one of the most important counter-intuitive examples in the study of mathematics. In the harmonic series, the numbers or terms get closer and closer to zero, while the series itself diverges.
7. Taylor Series: Definition, Formula & Examples
In mathematics, we sometimes gain insights by expressing functions in different ways. One of those ways is the Taylor series. In this lesson, we will define the Taylor series and use examples to demonstrate its use.
8. Taylor Series for ln(1+x): How-to & Steps
In this lesson, we show how to find the Taylor series for ln(1+x). This is a series evaluated about a specific point so we also look at the interval of convergence about this point.
9. Maclaurin Series: Definition, Formula & Examples
In this video lesson, you will learn how the Maclaurin series is a special case of the Taylor series. You'll also discover what some common Maclaurin series are for functions such as e^x and sin x.
10. Maclaurin Series for ln(1+x): How-to & Steps
In this lesson, we show how to find the Maclaurin series for a particular function: ln(1 + x). In addition to the steps for finding this series, you will also learn how to determine its region of convergence.
11. How to Apply the Ratio Test for Convergence & Divergence
Watch this video lesson, and you'll learn how you can use the ratio test to help you determine whether a particular series converges or diverges. You will also learn what to watch out for when using the ratio test.
12. Testing for Convergence & Divergence by Comparing Series
Comparing a series that you are given to a series that you know the answer to can help you to answer questions quickly and easily. Learn how to do so for convergence and divergence testing.
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Other chapters within the Math 104: Calculus course
- Graphing and Functions
- Vectors in Calculus
- Geometry and Trigonometry
- How to Use a Scientific Calculator
- Rate of Change
- Calculating Derivatives and Derivative Rules
- Graphing Derivatives and L'Hopital's Rule
- Applications of Derivatives
- Area Under the Curve and Integrals
- Integration and Integration Techniques
- Integration Applications
- Differential Equations
- Studying for Math 104