In the former article, we looked at the Power Series. We saw how we could find the radius of convergence, and therefore also the interval of convergence, by using the root test. We also saw how we could check whether the endpoints were included or not in the interval. In this article, we will look at different calculations we can make with the power series and their impact on the radius and interval of convergence.
Integration and Derivation
In this section, we will check what impact the derivation or integration of a power series has on the radius of convergence. Let us first define it formally:
What does this say? It simply says that if we either differentiate or integrate our series, then the result will have the same radius of convergence! We are just assuming that R is above 0 since otherwise, our series is divergent.
Example of Derivation and Integration
Let us take an example. Imagine that we consider a power series, which looks like the following:
And we have found out that it has the following radius and interval of convergence:
Let us then try to show that its derivative has the same radius and interval of convergence. Let us first try to find the derivative:
