College Math Teaching

May 29, 2013

Thoughts about Formal Laurent series and non-standard equivalence classes

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: $\sum^{\infty}_{j = -\infty} a_j t^j$; examples might be: $...-2t^{-2} + -1t^{-1} + 0 + t + 2t^2 ... = \sum^{\infty}_{j = -\infty} j t^j$. I’ll denote these series by $p(t)$.

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 $p(t)$ of “bounded power” if there exists some integer $M$ such that $a_m = 0$ for all $m \ge M$; that is, $p(t) = \sum^{k}_{j = -\infty} j t^j$ for some $k \le M$.

Equivalence classes: two Laurent series $p(t), q(t)$ will be called equivalent if there exists an integer (possibly negative or zero) $k$ such that $t^k p(t) = q(t)$. 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 $p_1(t), p_2(t), p_3(t)....p_k(t)....$ be a sequence of equivalent Laurent series. We say that the sequence $p_n(t)$ converges to a Laurent series $p(t)$ if for every positive integer $M$ we can find an integer $n$ such that for all $k \ge n$, $p(t) - p_k = t^M \sum^{\infty}_{j=1} a_j t^j$; that is, the difference is a non-Laurent series whose smallest power becomes arbitrarily large as the sequence of Laurent series gets large.

Example: $p_k(t) = \sum^{k}_{j = -\infty} t^j$ converges to $p(t) = \sum^{\infty}_{j = -\infty} t^j$.

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.