Formal series

I intend to write a series of posts on some topics I found often to be vaguely or mis-represented in baccalaureate's math courses. This one's about something called "formal series" in discrete mathematics. It's written in a rough and inaccurate manner because the true reason I'm publishing it now is that I'm afraid to forget about it again. I'll (re)write this whole thing later when I have time. Yet here's the gist (a bit more under the cut): don't try to fool students and call it "application of complex analysis to performing some operations on sequences".

These dudes introduce some magical "formal series" without specifying what they really are and what sequences are and why all of this really works. The way it oughts to be taught is as follows:

1. Since the very first semester (or even better in school) the folks should learn some (very basic) basics of functional analysis. Sets and functions. And the difference between the a funciton and its value at some point
2. Then you teach them that a sequence in the set $X$ is a map from $\mathbb{N}$ to $X$.
3. You teach them that a series in the v.s. $X$ is essentialy a sequence of partial sums and the series and and sequences in v.s. are interchangeable
4. You teach them about differentiability and power series
5. If your for your sequence $a$ the value $\limsup_k\sqrt[k]{|a_k|}$ is finite then the corresponding power series $\sum_k a_k (x - x_0)^k$ is convergent in some disc. That's right, it's Hadamard's bit, works well in Banach spaces
6. If you add one series to another (or multiple by a constant or do you name what to it) then it's going to just change the radius of convergence somehow. But the series will still converge at least somewhere. Well actually I'm not sure right now what happens if they're centered around different points I'll look through it after I finally get some sleep.
7. It's an (absolutely) convergent powerseries thus it defines a differentiable function!
8. You could try and mix and split your series into some parts resembling Taylor series of some well-known functions
9. How now, you've got combinations of some trivia? I reckon you know something about adding $\frac{1}{1-ax}$ to $\frac{2}{1-bx}$.
10. Mix and expand!