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Let $f:B^{m}\to\mathbb{C}$ be a multivariate function defined by a power series $f(x)=\sum_{\alpha}c_{\alpha}x^{\alpha},\,\,x\in B^{m}$, where $\alpha\in\mathbb{N}^{m+1}$ and $B^{m}=\{x\in\mathbb{R}^{m+1},\,\,||x||<1\}$. If $\sum_{\alpha}c_{\alpha}x^{\alpha}<\infty,\,\forall\,\,x\in B^{m}$, it is true that for each $p\in B^{m}$ there exist $\{d_{\alpha}\}\subset \mathbb{C}$ and a neighborhood $p\in B^{m}$ such that

$f(x)=\sum_{\alpha}d_{\alpha}(x-p)^{\alpha}$

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This is true. One approach is to extend the definition of $f$ to the complex unit ball (where the given series converges as well), use the multivariate Cauchy integral formula, and expand its kernel into a power series as in the one variable case.

For a purely real-variable proof, one can try to show that the $k$th derivative of $f$ at $p$ is bounded by $C^{|k|}k!$ for some constant $C$. But the estimates appear to be cumbersome in several variables.