Abstract

The author has often encountered difficulties making students grasp some of the less evident characteristics of the Taylor power series. In particular, it is sometimes difficult to make it clear that a function f(x) may, at a given point of its domain, possess derivatives of all orders and give rise to a convergent formal Taylor series, and yet the limit of that series need not coincide with f(x). Since a simple example is worth tens of pages, I have thought up one and present it in the first part of this educational Note. The Note then proceeds with the analysis of some not-so-trivial aspects of this special case.