So what's the basic idea behind Taylor series? (A "proof sketch", or even a sequence of theorems relying on each other building up to the lightbulb step.)
One of the things that was used less and less time was spent on, but they're convergent sum of a series that eventually approximates some function's value with increasing accuracy, from what I remember. You can get a sine value from one. Actually proving them, I can't remember how, but the class treated it pretty much as a short bonus topic.
(1) Rolle's theorem says that if f(x) if differentiable in the interval (a,b) and f(a)=f(b), there must be a point c such that f'(c) = 0.
Proving this one uses maths that, although simple, more analysis-like than calculus-like, but it should be intuitive if you try to draw a function with f(a)=f(b). It has to come up and down so it has a point where it has a horizontal tangent.
(2) The Mean Value Theorem is like Rolle's theorem, but "tilting" or "rotating" your function, so that if f(b) = r(f(b)-f(a)) then there's a point c with f'(c) = r. And this is already an order 1 Taylor series! It gives
f(b) = f(a) + (b-a) f'(c) for some c
(3) Now what remains is just induction. For an order-2 series, write the order-1 series for f'(c) as a function of f'(b). You'll start to see the pattern. If you really want to put a bow on it, you can assume the order (n-1) series exists with the known formula and prove the order (n) series works out.
