I’m writing this post as a way of preparing for a lecture. I want to discuss the result that a power series is differentiable inside its circle of convergence, and the derivative is given by the obvious formula . In other words, inside the circle of convergence we can think of a power series as like a polynomial of degree for the purposes of differentiation.
A preliminary question about this is why it is not more or less obvious. After all, writing , we have the following facts.