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?