Consider the function $f$ on $\mathbb{R}^{l}$ given by \begin{eqnarray}\left(x\_{1},...,x\_{l}\right)\mapsto\left(\sum\_{i=1}^{l}\frac{1}{\left(1+x\_{i}\right)^{k\_{i}}}-\left(l-1\right)\right)^{-1} \end{eqnarray} where $l\geq1$ and $k\_{1},...,k\_{l}\in\mathbb{N}$. I claim that the corresponding (multivariate) Taylor series around $x=0$ converges for all $x\in\mathbb{R}\_{+}^{l}$ with $\sum\_{m=1}^{l}\frac{1}{\left(1+x\_{m}\right)^{k\_{m}}}>l-1$. At first glance this looks obvious, but (at least for me) it isn't. Does anybody have an idea on how to show this?



Radius of convergence of multivariate Taylor series
Let $x = (x\_1,\ldots,x\_l)$. We can write the function $f(x)$ defined in the question as $f(x) = P(x)/(1 - Q(x))$, where $P$ and $Q$ are polynomials and $Q$ has *non-negative* coefficients. Therefore, for $x \in \mathbb{R}\_+^l$ such that $Q(x) < 1$, we can expand and rearrange the terms of the series $\sum\_{n = 0}^\infty (Q(x))^n$ (and still get a convergent series) to get the Taylor series of $1 / (1 - Q(x))$. Now we multiply this series by $P(x)$, expand and rearrange the terms once again, and we arrive at the Taylor series of $P(x) / (1 - Q(x))$, which therefore converges in the region $\{x \in \mathbb{R}\_+^l : Q(x) < 1\}$. But this is precisely the region the original question asks for.


