> which you can easily verify is its own derivative simply by differentiating it term-by-term.
Yes, but it's not a priori clear that the series actually even converges for all x, and the fact that power series are differentiable term-by-term within their radius of convergence also requires proof.
Fair enough, but IIRC students are first taught about converging and non converging series in pre calc. So then you get to calc, and work on the e proof, but first you say, "if you paid attention last year, you'll remember this series converges."
Mathematics education seems to be really different depending on the country (I'm in Europe). I barely saw a proof (except maybe some geometry proofs) in any of my high school maths classes and had to learn what a mathematical proof is in university. But I think this also depends on what focus area / profile you have in high school (mine was languages).
Yes, but it's not a priori clear that the series actually even converges for all x, and the fact that power series are differentiable term-by-term within their radius of convergence also requires proof.