Power Series - Representation of Functions - Calculus 2

So far we have seen that power series create functions that are defined for a domain of x values for which the power series is convergent. However, can we actually take a specific function and express it as a power series? Well, there's actually several methods for rewriting functions as power series, but the first way we will focus on, is when the function of interest (typically a rational function) is in a form similar to that of the sum of a geometric series.

When a rational function is similar to the form of the sum of geometric series, what we can do is algebraically manipulate the function until it is in the form of a geometric sum. This includes having a common ratio r, which needs to involve the variable x in some way, depending on where we want the power series representation to be centered. If we want the series to be centered at c = 0, then the common ratio just needs to involve x in some way, but if we want the series to be centered at any other value, then the common ratio needs to involve the quantity (x-c) instead. Either way, once the function is in the form of a geometric sum, it will be equal to a geometric series whose common ratio involves the variable x, which would make it a power series.

Determining how to manipulate a rational function into the proper form such that it can be rewritten as a (geometric) power series is the heart of this lesson, but we will also look at determining the radius and interval of convergence for this resulting power series as well. So with that, let's get to it!

In this lesson, you will learn:

• How to manipulate a function similar to the form of the sum of a geometric series to be in that desired form
• How to rewrite a function in the form of a geometric sum as a power series
• How to find the radius and interval of convergence for the power series that represents the function

I hope you find this video to be helpful!

-Josh