So the thought goes: suppose you'd like a function that looks like 1/x near x=0. I give you 1/x. Another satisfied customer! "No, no", you say, "I wasn't done ordering. I also want it to look like 1/x near x=1". I give you 1/x again. Bam! "No, I should have said, I want it to also blow up the way 1/x does at x=0, but to blow up the same way at x=1". Ok, so you want it to look like

*1/(x-1)*does near x=1. Fine. I give you 1/x + 1/(x-1). This blows up at x=0 and x=1. You hum to yourself, somewhat more satisfied, but say you want it to blow up at every integer. I try offering

Sum[1/(x-n), n = -∞...∞]

...but it doesn't converge anywhere! Harmonic series (or, you know, series that are 'close enough', i.e. bounded below in magnitude by harmonic series) are like that.

And so the way I like to think about the 'renormalization' trick here is to:

(1) differentiate the function in question

(2) do the infinite sum

(3) integrate

and this works!

(1) The derivative of 1/(x-n) is -1/(x-n)^2.

(2) Wolfram alpha tells me that

Sum[-1/(x-n)^2, {n, -Infty, Infty}] = -(ψ'(-x)+ψ'(1+x))

where ψ is the digamma function, which is defined to be Γ'/Γ where Γ is Euler's continuous generalization of the factorial. (well, off by one, for some reason that's always escaped me)

(3) the integral of that is ψ(-x) - ψ(1+x) + C, which Wikipedia tells me is equal to -π cot(-π x) + C.

So there it is! I tricksily added an infinite series of functions that didn't converge and got an answer anyway somehow --- a nice simple trigonometric one, in fact --- albeit one with an arbitrary constant C out back, whose value is arguably infinite somehow, maybe?

I've always been struck by how much grumbling I hear about the weirdness of renormalization, but if this kind of trick is the essence of it, (and any physicists in the audience feel free to tell me it's not if it's not) then it doesn't bother me a bit. It's just telling me that what's 'real' (I'm tempted to say: what's 'physical') in this situation is the derivative -1/(x-n)^2 rather than the original function 1/(x-n).

And that's fine! Physics is always telling us things like this, that energy values are not real, energy

*differences*are. Phase isn't real, phase

*differences*are, etc. etc. Small wonder that if we add an infinite number of copies of our conventional 'energy offset' that it blows up to infinity. Don't do that, then!