Alternating Series Test & Remainder - Calculus 2

Another special type of series you may encounter is what is known as an alternating series. An alternating series is a series whose terms alternate between being positive and negative. This is usually caused by having the expression (-1)^n or (-1)^(n+1) within a series. In either case, both expressions cause the terms of a series to switch back and forth between being positive and negative with each consecutive term.

In order to determine if an alternating series converges or diverges, we can use a special test known as the Alternating Series Test. This test has two requirements to check that allow us to conclude that an alternating series converges. First, the series needs to fail the divergence test, and the sequence from the series (excluding the part that makes it alternating) needs to be decreasing. If these two requirements are met, we know that the alternating series must converge. 

If an alternating series is convergent, unlike some other types of series, it is possible to approximate the sum of the series. Due to the unique properties of a converging alternating series, a special error formula comparing the actual sum and the nth partial sum of the series can be used to determine what specific partial sum can be calculated to accurately approximate the sum within a certain margin of error. That may sound a little complicated, but once you see this process in action it should make a lot more sense. And so let's not waste anymore time and get right to it!

In this lesson, you will learn:

• What an alternating series is and how to identify one
• How to use the Alternating Series Test and why it works
• How to approximate the sum of an alternating series using the error formula

I hope you find this video to be helpful!

-Josh