Differentiating and Integrating a power series

 Objectives:

1) Differentiate power series term by term

2) Integrate power series term by term

Let's f be the function that represents this power series. We can write:

The function f being a polynomial, we can differentiate and integrate it by differentiating and integrating each term of the polynomial in the same way we do it for a regular polynomial:

Differentiating f we have:

Integrating f we have:

Both the derivative and the integral of f converge on the interval I. Differentiating and integrating the function f this way is called differentiating and integrating term by term the power series that represents the function f .This property allows us to do two things. First, knowing the power series representation of a function f, we can find the power series representation of its derivative. Also knowing the power series representation of a function f allows us to find the power series representation of its integral. For example if I know the power series representation of  the function f(x) = 1/x-1, I can  differentiate term by term find the power series representation of g(x) = 1/(x-1)², which is the derivative of f . Similarly, knowing the power series representation of f(x) = 1/1+x, I can integrate term by term to find the power series representation g(x) = ln(1+x), which is the integral of f(x).

Theorem 

Example

Solution

Practice