You gave two scenarios which are worth addressing:
"x-x=0". Here we have two terms (x and -x). Their magnitude grows at the same rate, so it is not surprising that their difference is fixed and the two-term "series" "converges" to zero.
"Any converging series." Consider the exponential growth function: e^x = 1 + x + x^2/2 + x^3/6 ... + x^n/n!. This grows fast, but we know that the factorial is super-exponential so the terms must approach zero for any fixed x. If the series converges (true though not obvious) then the value must increase very quickly.
Now consider the exponential decay function. Unlike x-x, the terms are raised to different powers, so they grow at different rates. And unlike the exponential growth function, the series converges to zero in the limit.
The "miracle" is not the convergence but how it happens: terms growing in opposite directions at wildly different rates. It's as if a lopsided spaceship chaotically fired rockets at full power in all directions, and happened to stay exactly still.
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.
"x-x=0". Here we have two terms (x and -x). Their magnitude grows at the same rate, so it is not surprising that their difference is fixed and the two-term "series" "converges" to zero.
"Any converging series." Consider the exponential growth function: e^x = 1 + x + x^2/2 + x^3/6 ... + x^n/n!. This grows fast, but we know that the factorial is super-exponential so the terms must approach zero for any fixed x. If the series converges (true though not obvious) then the value must increase very quickly.
Now consider the exponential decay function. Unlike x-x, the terms are raised to different powers, so they grow at different rates. And unlike the exponential growth function, the series converges to zero in the limit.
The "miracle" is not the convergence but how it happens: terms growing in opposite directions at wildly different rates. It's as if a lopsided spaceship chaotically fired rockets at full power in all directions, and happened to stay exactly still.