Given the function $$ f(x,y,z) = \frac{1}{x^2+y^2+z^2} \qquad x,y,z\in \mathbb{R} \;, $$ I want to calculate the domain of convergence of the corresponding Taylor expansion at point $P_0 = (x_0, y_0, z_0).$ How can that be accomplished?
If the function depends only on a single variable that is done by the Cauchy-Hadamard theorem. But how do you do that if it depends on multiple variables? I found this journal article about convergence of power series in several complex variables and "Reinhard domains" https://arxiv.org/abs/1601.00274 helpful for understanding but I am still not able to calculate the domain of convergence.
The only thing I was able to calculate up to now by a numerical approximation is the convergence radius of the Taylor expansion of $f(x,y,z)$ at $P_0$ in the direction pointing towards the singularity. The result (as expected) is $\sqrt{x_0^2+y_0^2+z_0^2}$ for that path. But what is the convergence radius if I choose a path that is passing by the singularity at a certain distance and not passing through the singularity? Is it infinite?