Comment by antognini | Hacker News Reader

Comment by antognini | Hacker News Reader

One of the things that has always seemed rather magical to me is the Taylor series of the exponential function for very large negative values. We know that a number like e^-100 is a number extremely close to zero. Yet when you write out the Taylor series you see an alternating some of increasingly large, seemingly arbitrary numbers (at least for ~100 terms):

e^-100 = 1 - 100 + 5000 - 166,666.6 + 4,166,666.6 - ...

If you were just given this sum and knew nothing about Taylor series or the exponential function you'd assume that its value was some extremely large number. Yet everything manages to cancel out just right so that the resulting sum is almost exactly zero, but not quite.

I can't help but wonder if there's some parallel to parts of quantum field theory. If you expand out the interactions between two particles you also end up with a series that is apparently divergent. Yet we know experimentally that the first few terms work quite well as an approximation. It feels a bit like you're looking at the series of e^-100 without knowing about the exponential function.

The whole polynomial functions that give the truncations of the exponential Taylor series are also well-behaved in a way that's really surprising if you haven't thought about it before. For example, if you plot 1, 1 + x + x^2/2, 1 + x + x^2/2 + x^3/6 + x^4/24, and so on, you'll see that they actually don't oscillate at all as you might expect for negative inputs (for the reason you're saying), but rather they're very nice convex "bowl-shaped" positive functions, which hug zero more and more closely along the negative axis as you include more terms. One way to get some intuition about this is to notice that the derivatives of the next polynomial in the sequence are just the ones that came before (related to exp(x) being its own derivative). In particular, if all the even degree ones so far have been convex, then so will the next one be.

The idea you mention at the end is not specific to something mathematically advanced like quantum field theory. Useful truncations of divergent series show up in many more elementary situations in calculus and applied math; here is a nice class taught by the excellent Steven Strogatz where he talks about a simple example in the first lecture:

https://www.youtube.com/watch?v=KZsk8B_z8pI&list=PL5EH0ZJ7V0...

I have been watching this class (to supplement a class on dynamically systems where we are learning perturbative methods). Stroganoff has a good text book, a good pop sci book, a really good podcast and now really good technical lectures. One small positive result of Covid.

Why do you think a series which absolutely converges but whose first few terms are a poor approximation is related to a series which diverges but whose first few terms match the empirically expected value?

Not only is it a stretch to wonder about parallels between a basic Calc II concept and Quantum Field Theory, but is seems like the exponential function is the exact opposite of the example you provided.

I was being a bit loose in my language, but I was more specifically thinking of the connection to "resurgence theory" which aims at understanding more complicated kinds of apparently divergent infinite sums. See this article for a popular overview: https://www.quantamagazine.org/alien-calculus-could-save-par...

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