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.

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