Power Series - Derivatives & Integrals - Calculus 2

Previously we established that a power series creates a function that is defined for the values of x for which the power series is convergent. Well since a power series is a function, we are naturally interested in knowing its derivative and integral (antiderivative). To find each of these functions, we need to know the rules for differentiating and integrating a power series.

Fortunately, finding the derivative and integral of a power series is not all that difficult. We could start out by individually differentiating/integrating each term of the power series, but we would begin to find that since each term of a power series just involves x raised to some power, the differentiation/integration process only involves using the power rule for derivatives/integration. As a result of this, we don't need to individually differentiate/integrate each term of a series to find the derivative/integral. Instead, we can simply use the power rule for derivatives/integration on the nth term of the power series and be done in just a couple seconds. However, finding the derivative and integral of a power series is only half of the work that needs to be done.

Similar to the original power series that represents f(x), the derivative and integral of a power series will only be defined for an interval of x values for which the resulting power series is convergent. As such, they will also have intervals of convergence that will need to be found. Conveniently, the derivative and integral of the power series will share the same radius of convergence and interval of convergence, but the endpoints of the interval of convergence may slightly differ. Specifically, the derivative and integral may lose or gain endpoints for their interval of convergence compared to the original series (closed brackets vs open brackets). Because of this, we will need to manually check the endpoints for the interval of convergence for all three functions: the original power series, its derivative, and its integral. We go through this process and explain each step along the way in this new lesson, so lets get right into it!

In this lesson, you will learn:

• How to differentiate a power series
• How to integrate a power series
• How the radius and interval of convergence for a derivative and integral compare to the original power series

I hope you find this video to be helpful!

-Josh