Power series interval of convergence, why root test works?

Question

For a given power series $$S = \sum_{n}c_n (x - a)^n$$ We have interval of convergence for it defined as the set of values of $x$ for which the series converges.

Many reference books says I can use ratio test or root test for the radius, e.g. $$\limsup_{n\rightarrow+\infty}\sqrt[n]{\left | c_n (x - a)^n\right |} = \limsup_{n\rightarrow+\infty}\sqrt[n]{\left | c_n \right |}\left | x - a\right |< 1$$

Root test is a sufficient condition for convergence but never a necessary condition. If we use root test to find the interval of convergence, we'll arrive at a subset of the true interval.

And books give that we should check the endpoints of the interval. I could understand that if the series diverges on both endpoints, then the values outside the interval just diverges as well. But what if the series does converge on either endpoint or both? How do we confirm that there are no possible value outside the interval that still makes the series converge?

Answer 1

Finally, I think I get the answer. (The key part is that root and ratio test are for both convergence and divergence.)

The root or ratio test in this process actually gives three part of results.

For some $x$, the limit is greater than 1, and the series diverges.

For some other $x$, the limit is less than 1, the series converges.

For the rest $x$, the result is unknown, and is left as endpoint. (Provided that the function in the limit is continuous)

So that only endpoint are left in undefined state that requires further work and which is tested with other methods.