Fix (open) polydisks $B' \subset B \subset \mathbb{C}^n$ and $\epsilon >0$. If $f$ is holomorphic on $B$, then there exists a polynomial $P$ such that $$\sup_{z \in\ B'}|f(z)-P(z)|<\epsilon.$$
Can we prove this by truncating the power series of $f$?
This result is used to prove Grothendieck-Poincare' lemma in Corollary 1.3.9 of the book Complex Geometry by Huybrechts
We need that $B' \subset\mspace{-2mu}\subset B$ of course, otherwise such an approximation is generally impossible. If we have the assumption that $B'$ is relatively compact in $B$, the assertion is a direct consequence of the locally uniform convergence of the power series of $f$ (centered at the centre of $B$) on $B$.
Assume for simplicity that the centre is $0$, then pick a point $\mathfrak{z} \in B$ such that for all $1 \leqslant k \leqslant n$ $\lvert z_k\rvert > \sup \{ \lvert w_k\rvert : \mathfrak{w} \in B'\}$. Then, since the power series converges absolutely in $\mathfrak{z}$, we have
$$\lvert f(\mathfrak{w}) - P_N(\mathfrak{w})\rvert = \left\lvert \sum_{\lvert\nu\rvert > N} a_\nu \mathfrak{w}^\nu\right\rvert \leqslant \sum_{\lvert\nu\rvert > N} \lvert a_\nu\rvert \, \lvert \mathfrak{z}\rvert^\nu \xrightarrow{N\to\infty} 0$$
for all $\mathfrak{w} \in B'$, where $\lvert\mathfrak{z}\rvert = (\lvert z_1\rvert,\dotsc,\lvert z_n\rvert)$, $\lvert \nu\rvert = \nu_1 + \dotsc + \nu_n$, and $\mathfrak{w}^{\nu} = w_1^{\nu_1}\cdot \dotsc \cdot w_n^{\nu_n}$ as usual for power series in several variables.