Nth Derivative

This lecture focuses on studying the n-th derivative of a function defined by a power series with a positive convergence radius or infinite radius, showing how to obtain the n-th derivatives through a recursive process. It also demonstrates that functions defined by power series are infinitely differentiable, leading to the conclusion that they are very regular functions. Additionally, it introduces the concept of C omega functions, which are even more regular than C infinite functions, and discusses evaluating the n-th derivative at a specific point using Taylor's formula. The lecture concludes by exploring functions represented by power series and their relationship with known functions.