I have shown that the $\sum_{n=0}^{\infty}\frac{n}{3^n+z^n}$ converges locally uniformly on $D=\{z\in\mathbb{C} : |z|<3\}$ but I am struggling to show that it is not uniformly convergent on this region. I suspect that problems arise around $z=-3$ but since $-3$ isn't included in the region $D$ I don't know how to go about this.
The only test I am aware of (to show $f_n$ does not converge uniformly to $f$) is showing that there is some sequence $x_n\in D$ such that $|f_n(x_n)-f(x_n)|\geq c$ for some constant $c>0$. I have tried doing a similar thing here with the sum, assuming that it converges to some limit function $F$ but it doesn't seem to get me anywhere.