Owl: A power series looks like an infinite polynomial: Σ cₙ(x − a)ⁿ. Inside a certain radius it behaves politely — you can differentiate and integrate term by term with care. Outside that radius, it falls apart.
Think of a flashlight beam that only stays focused on part of the number line. The radius R marks how far from the center a the series "works." Endpoints of the interval need their own checks — they are not automatic.
We'll find R with the ratio test, test endpoints, and remember: power series do not converge everywhere by default, and skipping endpoints is a classic miss.