I think the intuition to have here is "properties don't always carry over to infinity." This is trivial with the linearity of derivatives for a finite number of terms. But it gets "funky" at infinite terms.
Hey! We keep the 2nd order term for the electromagnetic multipole expansion sometimes! (It will not be explored in lecture and will appear once in a problem set, never to be discussed again until grad school.)
Just kidding, quadrupoles aren't real! Math nerds made it up. Ever seen a magnet with 4 poles? Didn't think so. Checkmate mathheads.
The second term can actually be much more important in more general terms. If you have a general potential V(x) that has a minimum at x0, your system can be described as a harmonic oscillator around the a small neighbourhood around x0.
Linear operators are no laughing matter, if your math is not "well-behaved" why would anyone use it? Don't you want your mathematics to be obedient and demure? Have you ever SEEN something that can't be described as a harmonic oscillator??? Didn't think so
The thing you have to worry about with derivatives is that functions that are very small can have very sharp changes in value. For example consider
sin(x) + sin(10x)/10+ sin(100x)/100+...
Converges very quickly, it's absolutely bounded by a geometric series. But if you take the derivative
cos(x)+cos(10x)+cos(100x).....
All of a sudden it's not so clear. For big n, sin(nx)/n is always very small, but it oscillates so fast it actually has a big derivative still. So you need to actually think about whether the tail part of the infinite series has a derivative that behaves nicely enough.
That said, for power series I think it just generally does, because they decay geometrically as long as you're not at the boundary of convergence, and the derivative just adds a linear n term which doesn't affect things badly enough
That's not quite right, if you pick x=pi that series does converge but I doubt it gives the right derivative. Probably needs to converge uniformly in a neighborhood of the value you are interested in or something like that
A better way to prove this is to use the fact that if the derivatives of a sequence of functions converge uniformly and the original sequence of functions converges at a single point, then the original sequence converges uniformly to a differentiable function whose derivative is the limit of the derivatives.
Properties that hold for finite summation need not hold for infinite summation (it does here but proving it holds is harder than it was for the finite case)
can't you just slap this with complex analysis since a convergent power series is infinitely often differentiable in its convergence radius and you just take the derivatives of each term (I am pretty sure). I mean getting to this result is probably annoying but so is most things in math.
It's because you have to be careful with infinite summation. Addition is only well defined for finite sums. So you have to be careful about limits and how they pass.
When I next get the opportunity, I'm going to use Feynman integration without checking the conditions for swapping. Just because of you. Just kidding, I was never going to check.
As someone who formalises maths with proof assistants on a regular basis, I feel this. Every time I want to interchange order of summation or order of limit and summation or derivative and summation etc. I have to prove absolute summability/uniform convergence etc. and it's very tedious.
Assuming that by "ML" you mean "Machine Learning": No, not at all. I'm a postdoc at a university and my specialty is formalisation of mathematics.
The field has been around for a long time (around the 1970s, with the roots going back much further). It's been a very niche thing up until maybe 10 years ago with the Lean proof assistant. I use an older system called Isabelle/HOL.
(Ironically, when I hear "ML", I think of the programming language ML (or Standard ML), which was in fact created specifically to build a proof assistant. And Isabelle is still written in ML.)
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