TITLE: To what extent is the taylor polynomial the best polynomial approximation?
QUESTION [29 upvotes]: Given a function $f\in\mathscr C^n([a,b])$ and a point $x_0\in [a,b]$, to what extent is the n-th taylor polynomial $T_n(x,x_0)=\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$ the best polynomial approximation of $f$ in $[a,b]$ ? This may seem to be dumb question, but is there a metric $\rho$ on $ C^n([a,b])$ so that $\rho(T_n(x,x_0),f)=\min\{\rho(p,f)\mid \text{p is a polynomial function} \}$ ? 
Thank you

REPLY [6 votes]: $T_n(x, x_0)$ is the only polynomial of degree less than or equal to $n$ such that
$$ T_n(x, x_0) - f(x) \in o((x - x_0)^n) $$
or in terms of limits,
$$ \lim_{x \to x_0} \frac{T_n(x, x_0) - f(x)}{(x - x_0)^n} = 0$$