[Tainnor]

> 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.

[jgalt212]

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."

[Tainnor]

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).