I see, it's more amazing than my simple examples. But why is it more amazing as this presumably uninteresting series I just made up
sum(n=1...infinity, 1/2^n - 1/(2^n+1))
That also has alternating terms that grow differently in opposite directions and have huge numbers, yet they cancel out so that it converges to something. Is that uninteresting because each term in a pair has the same exponent, or because it doesn't converge to something that's closer to zero as something in the terms increases?
Maybe you can't easily just make up a converging series that has all the features they listed and that uniqueness makes it interesting?
I would say it's not as interesting because a high-schooler (or Pythagoras) could easily verify the fact AND they would easily be able to explain how you came up with that sum.
One of my metrics for determining if a fact F is interesting is if P(F) is much larger than V(F).
Here P(F) is, vaguely speaking, the amount of effort required to "explain" or "generalize" the fact, and V(F) is the amount of effort required to verify the fact. In the case of e^{-x}, each individual example can be verified by an elementary schooler so V(F) is very small.
On the other hand,
I don't see how you could explain the phenomena of convergence to 0 without teaching the elementary schooler a large part of calculus, so P(F)/V(F) is large.
sum(n=1...infinity, 1/2^n - 1/(2^n+1))
That also has alternating terms that grow differently in opposite directions and have huge numbers, yet they cancel out so that it converges to something. Is that uninteresting because each term in a pair has the same exponent, or because it doesn't converge to something that's closer to zero as something in the terms increases?
Maybe you can't easily just make up a converging series that has all the features they listed and that uniqueness makes it interesting?