College Math Teaching

January 20, 2014

A bit more prior to admin BS

One thing that surprised me about the professor’s job (at a non-research intensive school; we have a modest but real research requirement, but mostly we teach): I never knew how much time I’d spend doing tasks that have nothing to do with teaching and scholarship. Groan….how much of this do I tell our applicants that arrive on campus to interview? 🙂

But there is something mathematical that I want to talk about; it is a follow up to this post. It has to do with what string theorist tell us: \sum^{\infty}_{k = 1} k = -\frac{1}{12} . Needless to say, they are using a non-standard definition of “value of a series”.

Where I think the problem is: when we hear “series” we think of something related to the usual process of addition. Clearly, this non-standard assignment doesn’t related to addition in the way we usually think about it.

So, it might make more sense to think of a “generalized series” as a map from the set of sequences of real numbers (or: the infinite dimensional real vector space) to the real numbers; the usual “limit of partial sums” definition has some nice properties with respect to sequence addition, scalar multiplication and with respect to a “shift operation” and addition, provided we restrict ourselves to a suitable collection of sequences (say, those whose traditional sum of components are absolutely convergent).

So, this “non-standard sum” can be thought of as a map f:V \rightarrow R^1 where f(\{1, 2, 3, 4, 5,....\}) \rightarrow -\frac{1}{12} . That is a bit less offensive than calling it a “sum”. 🙂


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