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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.)
by thanatropism 10y ago
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.)
- Chris2048 10y agoIf OP doesn't reply, can you? I'd like any insight I can :-)
- thanatropism 10y ago(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.
- jimmaswell 10y agoOne 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.