Convergence of a series of holomorphic functions

Question

A frequent type of exercise, in my complex analysis course, consists of determine where a series of holomorphic functions converges to a holomorphic function. For example: $$f (z)=\sum_{n=0}^\infty\frac {z^2} {z^3-n^3}.$$ I would say that this function is mermorphic with simple poles in the points $n, ne^{\frac 2 3 \pi i}$ and $ ne^{\frac 4 3 \pi i}$, because it's where the sum doesn't converge (and if $f$ goes to infinity in $z_0$, $z_0$ is a pole). However I don't know if it suffices to prove that the sum is finite in a $z_0$, in order to prove that $f $ is holomorphic there. A sequence of holomorphic functions, uniformly converging over any compact set, converges to a holomorphic function; can one say that the partial sums of a series always form a uniformly converging sequence, given that the series converges? This seems reasonable to me, but I'm not sure at all that is true. Thanks in advance

Answer 1

"can one say that the partial sums of a series always form a uniformly converging sequence, given that the series converges?"

The answer to the above is no, but counterexamples are subtle (either one uses deeper theorems like Runge but the counterexamples are non-constructive or one can give clever examples that exhibit for example entire functions $G$ that are bounded on any ray through zero and from there by taking $F(z)=\frac{G(z)-G(0)}{z^n}, n \ge 1$ order of $G-G(0)$ at zero, $F(0) \ne 0$ and $F(re^{i\theta}) \to 0, r \to \infty, \theta$ fixed, so $f_n(z)=F(nz), n \ge 1$ cannot converge locally uniformly on a disc containing zero as $f_n(z) \to 0, z\ne 0, f_n(0)=F(0) \to F(0) \ne 0$, and taking $g_n(z)=f_{n}(z)-f_{n-1}(z), n \ge 1, f_0(z)=0$ we get a series counterxamples too)

However in the example in the OP and similar such one proves locally uniform convergence fairly easily using that if $z$ is in a compact set $K$ that avoids the poles (hence $|z| \le M$ for some positive $M$ by compactness), most terms $\frac{z^2}{z^3-n^3}$ are bounded in absolute values by say $\frac{8M^2}{7n^3}$, when $n \ge 2M$ so their sum is absolutely hence uniformly convergent on $K$ while the finitely many remaining terms do not matter as their finite sum is clearly a holomorphic function on $K$ anyway