Calculate the stalk of the direct image of the punctured affine space at origin 
Let $i$ be the inclusion of $\mathbb{A}^n\,\backslash\{0\}$ in $\mathbb{A}^n$, where $\mathbb{A}^n$ is the affine space over some algebraically closed field $k$. Determine the stalk of $i_*\mathcal{O}_{\mathbb{A}^n\,\backslash\{0\}}$ at $0$ for $n=1,2,\cdots$

This is an exercise problem for an Algebraic Geometry course. My solution is as follows and I am not sure if it is correct:

Let $U_j=\mathbb{A}^n_{x_j}=\operatorname{Spec}k[x_1,x_2,\cdots,x_n][x_j^{-1}]$. Then $\mathbb{A}^n\,\backslash\{0\}=\bigcup U_j$. So the stalk of $i_*\mathcal{O}_{\mathbb{A}^n\,\backslash\{0\}}$ at $0$ is the union of stalk of $(i|_{U_j})_*\mathcal{O}_{U_j}$ at $0$, which is the union of $S^{-1}k[x_1,x_2,\cdots,x_n][x_j^{-1}]$, where $$S=\{f\in k[x_1,x_2,\cdots,x_n]|f(0)\ne0\}.$$ Therefore the final result should be
$$\bigcup_{j=1}^n S^{-1}k[x_1,x_2,\cdots,x_n][x_j^{-1}].$$

Besides, I am not sure if the result could be simplified.
 A: Since $X := \mathbb{A}^n$ is affine, the global sections functor induces an equivalence of categories between quasicoherent sheaves on $X$ and modules over $\Gamma(X,\mathcal{O}_X) \simeq k[x_1,...,x_n]$. Now global sections of $i_*\mathcal{O}_U$ is just global sections of $\mathcal{O}_U$ over $U$, and so for $n=1$ we get the ring $k[x,x^{-1}]$. When $n > 1$, a standard calculation gives $\Gamma(U,\mathcal{O}_U) \simeq k[x_1,...,x_n]$. One can check that the $\Gamma(X,\mathcal{O}_X)$-module structure on the latter is the usual one.
As for stalks, recall that in the affine case we may compute the stalk of a quasicoherent sheaf as a localization of its module of global sections. For $n=1$, we therefore need to understand the localization $k[x,x^{-1}]_{(x)}$ of Laurent polynomials at the prime ideal $(x)$ (note we're inverting everything that's not a multiple of $x$). One can check that the result is $k(x)$, the field of rational functions in $x$ viewed as a module over $k[x]$. The case $n>1$ is straightforward.
