Absolute & Conditional Convergence - Calculus 2

Previously we saw that if a series has terms that alternate between being positive and negative that we could determine the convergence or divergence of that series by using the alternating series test. However, if we encounter a series that has both negative and positive terms, but does not alternate with each term, the alternating series test cannot be used. Instead, we will need another way to determine if such a series converges or diverges. 

This is where we introduce two new types of convergence known as absolute convergence and conditional convergence. Up to this point in Calculus 2, we have only referred to a converging series as just straight up convergent. However, when a series has both positive and negative terms (including alternating series), it is going to be important to analyze further and determine if that series converges absolutely or conditionally. In other words, does the series converge no matter the sign of the terms (absolute), or does it only converge when the terms are both positive and negative (conditional)?

We can test for these types of convergence by looking at a series where we have the absolute value of the sequence from the series. Depending on the convergence or divergence of this series, we can make conclusions about the convergence of the original series. Specifically, we will be able to say that it either converges absolutely, converges conditionally, or diverges. Determining which of these conclusions we can make for a given series will be the focus of this lesson. Let's get to it!

In this lesson, you will learn:

• What it means for a series to convergence absolutely or conditionally.
• How to test a series for absolute convergence.
• How to test a series for conditional convergence.

I hope you find this video to be helpful!

-Josh