Power Series - Representation By Derivatives & Integrals - Calculus 2

In the previous lesson we saw how functions that are similar to the form of the sum of a geometric series can be represented with a power series. However, we can extend this idea to more functions. Specifically, we can represent functions whose derivative or integral is similar to that form of a geometric sum as a power series as well. In order to do this, we will need to make use of our integration and derivative rules for power series, as well as the concept that integration and differentiation are opposite operations.

The overall process looks like this:

If we want to represent a function as a power series, and that function has a derivative similar to the form of a geometric sum, we can represent that derivative as a power series, and then integrate that power series to get back to the original function, and the resulting power series will represent that original function.

If we want to represent a function as a power series and that function has an integral (antiderivative) similar to the form of a geometric sum, we can represent that integral as a power series and then differentiate that power series to get back to the original function, and the resulting power series will represent the original function.

In both cases, we will then need to find the radius and interval of convergence for the power series that represents the original function, and that will be determined from the power series that represents the derivative or integral of the original function. All we have to do then is check the endpoints of the interval for the series that represents the original function. It may sound complicated with words, but the process is simpler than it sounds once you see it in action. So let's not waste any more time and get right into this lesson!

In this lesson, you will learn:

• How to integrate a power series to represent a function as a power series whose derivative is similar to the sum of a geometric series
• How to differentiate a power series to represent a function as a power series whose integral is similar to the sum of a geometric series
• How to find the radius and interval of convergence for the resulting power series in each case

I hope you find this video to be helpful!

-Josh