I have a rather radical idea. I start with Maclaurin Series! Let's see how it works. You first see graphically and "globally" that you get closer and closer to the function and when adding infinite terms you get the function. Then you have a point-wise look. For example, consider the Maclaurin series of Exp(x), you ask what happens at, say, x=1 (see the corresponding y-coordinates). Alongside "the convergent graphs", you have a numerical series. Playing with different functions, you get some interesting numerical series that without having any definition at hand it is not possible to decide whether they are convergent or not, ex. 1-1+1-1+1... It leads students to a definition of convergent numerical series and back again, convergent functional series! On the middle, we touch sequences.  

It seems strange, but usually I have a **big picture** for each of my courses and this idea works well within the picture I have for Single Variable Calculus.