1 min readβ’june 7, 2020

Athena_Codes

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. Letβs 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.

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