Approximation Rates for Multivariate Taylor Series

Let $k,n,m$ be positive integers and suppose that $f$ is $C^{k}(\mathbb{R}^n,\mathbb{R}^m)$ functions. For any given $\epsilon>0$ and $x_0\in \mathbb{R}^n$, are there known sharp approximation rates for $$ \inf_{\beta_{\cdot} \in \mathbb{R}}\,\sup_{\|x-x_0\|\leq \epsilon} \, \left\| f(x) - \sum_{|\alpha|\leq d} \beta x^{\alpha} \right\| \leq R(k,\epsilon,n,m,d) $$ where $R$ is independent of $f$? Herefore, I index $\beta$ depending on all multivariate monomials's exponents of degree at-most $d$