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:
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.
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...