About This Chapter
Series of Constants - Chapter Summary and Learning Objectives
This chapter dives into complicated series theories. Follow along with our expert instructors as they explore methods for testing limits to determine if they are convergent or divergent. Use the self-assessments to see how well you understand the key topics covered in the videos. The topics in this course include:
- How to use expanded form in math
- The definition of a harmonic series and its formula
- Understanding alternating series
- Identifying terms of series using rectangle areas
- How to test for convergence/divergence with the ratio test or comparing series
|What is Expanded Form in Math? - Definition & Examples||This video discusses how to deconstruct numbers into their component parts according to the value of individual digit places.|
|Harmonic Series in Math: Definition & Formula||Instructors discuss standard harmonic and alternating harmonic series, providing examples of each.|
|Alternating Series with Error Bound||Here you'll learn how to use the alternating series test to determine convergence.|
|Terms of Series as Areas of Rectangles||In this video the instructors describe the relationship of improper integrals to series as areas of rectangles and how to use the integral test for convergence.|
|How to Apply the Ratio Test for Convergence & Divergence||This lesson looks for absolute convergence in a series using the ratio test.|
|Testing for Convergence & Divergence by Comparing Series||Finally, the instructors describe how to use more than one series to test convergence through comparison.|
1. What is Expanded Form in Math? - Definition & Examples
In this lesson, we'll review place value and find out how it's useful when writing numbers in expanded form. We'll also look at a few examples of numbers written out in expanded form.
2. Convergence & Divergence of a Series: Definition & Examples
In this lesson, we motivate the concept of an infinite series by showing an example from basic physics. We use the example to introduce the geometric series and to further suggest the issues of convergence and divergence. This section is intended to be a preview of series in general. Convergence and divergence extends to all series outside the geometric type.
3. 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.
4. 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.
5. 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.
Earning College Credit
Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.
To learn more, visit our Earning Credit Page
Transferring credit to the school of your choice
Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.
Other chapters within the AP Calculus BC: Exam Prep course
- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Limits of Functions
- Function Continuity
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Parametric, Polar and Vector Functions
- Properties of Derivatives
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Derivative Applications
- Finding Derivatives
- Properties of Definite Integrals
- Applications of Integrals
- Using the Fundamental Theorem of Calculus
- Applying Integration Techniques
- Approximation of Definite Integrals
- Understanding Sequences & Series
- Taylor Series
- Using a Scientific Calculator for Calculus
- AP Calculus BC Flashcards