I've found the following exercise on 'Holomorphic functions and integral representation in several complex variables' of R.M. Range: prove that a power series $\sum c_\nu z_\nu$ and the derived series $\sum c_\nu (D^\alpha z^\nu)$ have equal domain of convergence for every multi-index $\alpha \in \mathbb N^n$.
For $n=1$ it is a standard result and I'm concerned about the case $n\ge 2$.
Let us consider $n=2$ and $c_{\nu_1,\nu_2} = \nu_2!$ if $\nu_1=0$, $c_{\nu_1,\nu_2}=\frac{1}{\nu_2!}$ if $\nu_1=1$ and $c_{\nu_1,\nu_2}=0$ if $\nu_1 \ne 0$ and $\nu_1 \ne 1$. If $\alpha=(1,0)$, then
$$ \sum_{\nu_1,\nu_2\ge 0} c_{\nu_1,\nu_2} D^\alpha(z_1^{\nu_1} z_2^{\nu_2}) = \sum_{\nu_1>0,\nu_2\ge 0} c_{\nu_1,\nu_2} \nu_1 z_1^{\nu_1-1}z_2^{\nu_2} = \sum_{\nu_2\ge 0} \frac{1}{\nu_2!}z_2^{\nu^2}. $$
So the domain of convergence of $\sum c_\nu (D^\alpha z^\nu)$ is $\mathbb C^2$. On the other hand,
$$ \sum_{\nu_1,\nu_2 \ge 0} |c_\nu z^\nu| = \sum_{\nu_2\ge0} \nu_2! |z_2|^{\nu_2} + \sum_{\nu_2\ge 0} \frac{1}{\nu_2!} |z_1||z_2|^{\nu_2} = +\infty, \mbox{for} \, z_1,z_2\ne0. $$
Is there something wrong with the statement of the exercise or am I missing something?