10.13 Radius and Interval of Convergence of Power Series

One of the questions we have about power series approximations of functions is where the approximation is valid, or in other words, where the power series converges. For a given x, we can find the radius, and then the interval of convergence for a power series.

For a Taylor series centered at x = a, the only place where we are sure that it converges for now is x = a, but we can expand this to a greater range using our knowledge of the ratio test. Let’s make an example to demonstrate this!

We have to test the endpoints by plugging them into x as sometimes it may converge or diverge at the endpoints.

Find the interval, radius of convergence, and the center of the interval of convergence.