Numberphile’s video on summing the positive integers has been confusing people, because physicists have a way of working with mathematics that is very foreign to mathematicians. Let’s see what Numberphile’s proposed proof can say about the sum of the positive integers that is mathematically rigorous.
We’ve been given
We can generalise this slightly
and for full generality we will treat these as formal power series. I’m not going to consider the functions they induce until right at the end.
Now, it’s perfectly mathematically rigorous to say that S_2(x) = 1 / (1 + x)^2 as formal power series (indeed it’s true when they are considered as functions on -1 < x < 1, but hold your horses!).
It’s also perfectly mathematically rigorous to say that
so that
which when the right hand side is expanded is
Note that in particular this is not S_2(x) = -3 S(x)! Now observe that if we consider the functions on R induced by these power series we do indeed find that
This is about as close as we can get, I think, using a rigourous form of Numberphile’s proof, to the physicists claim that
I have no doubt that when physicists use their identity, what they are doing is completely valid. What’s going on here is analogous to decategorification, that is, forgetting some important part of the structure. The important part of the structure here is the formal variable x that we hang coefficients onto to express the sum.
In the contexts that physicists use this identity, I assume that there is some other underlying structure which “decategorifies” to this sum of integers (perhaps it really is as simple as having this formal variable x). What’s wrong here mathematically is not the result but stating it and proving it in the “decategorified” setting, when the statement and proof are only valid in the original setting, not after “decategorification”.