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Hi everyone: Suppose $f$ and $g$ are two real analytic functions on a bounded domain $D$ of $\mathbb{R}^{n}$ and we have $$0\leq f(x)\leq g(x)$$ on $D$. Suppose the Taylor series associated to $g$ about $x_{0}$ is convergent for $|x-x_{0}|<r$. Can we conclude that the Taylor series associated to $f$ about the same point is also convergent on $|x-x_{0}|<r$?

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Hint: try $n=1$, $x_0 = 0$, $f(x) = 1/(1+x^2)$ and $g(x) = 1$.

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