I admit that I haven’t looked this up in the literature; I don’t know how much of this has been studied.
The objects of my concern: Laurent Series, which can be written like this: ; examples might be:
. I’ll denote these series by .
Note: in this note, I am not at all concerned about convergence; I am thinking formally.
The following terminology is non-standard: we’ll call a Laurent series of “bounded power” if there exists some integer such that for all ; that is, for some .
Equivalence classes: two Laurent series will be called equivalent if there exists an integer (possibly negative or zero) such that . The multiplication here is understood to be formal “term by term” multiplication.
Addition and subtraction of the Laurent series is the usual term by term operation.
Let be a sequence of equivalent Laurent series. We say that the sequence converges to a Laurent series if for every positive integer we can find an integer such that for all , ; that is, the difference is a non-Laurent series whose smallest power becomes arbitrarily large as the sequence of Laurent series gets large.
Example: converges to .
The question: given a Laurent series to be used as a limit, is there a sequence of equivalent “bounded power” Laurent series that converges to it?
If I can answer this question “yes”, I can prove a theorem in topology. 🙂
But I don’t know if this is even plausible or not.