What Is Degree N (Nth Order) Taylor Approximation? - Expii

The nth Taylor approximation of f(x) at a point x=a is a degree n polynomial, namely P(x) = ∑^(n)_(k=0) (f^k a)/(k!) (x-a)^k = f(a) + f'(a)(x-a)^1 + 1/2 f''(a)(x-a)^2 + ... + 1/(n!) f^n (a)(x-a)^n. This make sense, at least, if f is n-times-differentiable at x=a. The intuition is that f(a)=P(a), f'(a)=P'(a), f''(a)=P''(a), etc., up to f^n (a)=P^n (a): the "zeroth", first, second, etc., up to nth, derivatives match. (More concisely, f^k (a)=P^k (a) for 0 <= k <= n: the kth derivatives match for 0 <= k <= n.).