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?