Sequences of functions are pervasive in Mathematics and arise naturally in the process of approximating a given function by simpler ones, at least locally. This is for example the case when one looks at the Taylor polynomials of a given function f which is very regular in an open neighborhood I of a given point \(x_0\) , say for simplicity \(f\in C^\infty (I)\) . Upon computing higher and higher order derivatives of f at \(x_0\) one builds the sequence \((T_{n,x_0}f)_{n\ge 0}\) of the Taylor polynomials centered at \(x_0\) that are actually defined on the whole \(\mathbb {R}\) . It is natural to ask under what circumstances the numerical sequence obtained by evaluating each polynomial at one at the same point, say \(x\in I\) , converges to f(x). This example explains also how the idea of series of functions comes about, for each of the polynomials \(T_{n,x_0}f\) is constructed by summing powers so that again it is natural to ask what happens by fixing a point and considering the numerical series that arises. Both these examples will be discussed below but are just instances of a much more general picture.