Howtosignalprocessing
- How do I find the radius of convergence?
- What is the radius of convergence for the series?
- How do you test for convergence endpoints?
How do I find the radius of convergence?
The radius of convergence is half of the length of the interval of convergence. If the radius of convergence is R then the interval of convergence will include the open interval: (a − R, a + R). To find the radius of convergence, R, you use the Ratio Test.
What is the radius of convergence for the series?
The radius of convergence of a power series f centered on a point a is equal to the distance from a to the nearest point where f cannot be defined in a way that makes it holomorphic. The set of all points whose distance to a is strictly less than the radius of convergence is called the disk of convergence.
How do you test for convergence endpoints?
The endpoints of the interval of convergence must be checked separately, as the Root and Ratio Tests are inconclusive there (when x=±1L, the limit is 1). To check convergence at the endpoints, we put each endpoint in for x, giving us a normal series (no longer a power series) to consider.
