Let \(f(z)=\sum_{\nu\geq 0}a_{\nu}z^{\nu}\) be a holomorphic function in the open unit disk \(\mathbb{D}\) such that \(\limsup| a_{\nu}|^{1/{\nu}}=1.\) A point \(z_0\in\mathbb{C}\) is called a limit point of zeros of the partial sums \(S_n(f)(z)=\sum_{\nu=0}^na_{\nu}z^{\nu}\) if, for any \(\varepsilon>0,\) there exist infinitely many \(S_n(f)\) possessing a zero in \(\{z:\;| z-z_0|<\varepsilon\}.\) Let \(Z(S_n(f))\) be the set of all limit points of zeros of \(S_n(f)\). Jentzsch’s theorem ensures that the boundary of the disk of convergence \(\partial\mathbb{D}\) is a subset of \(Z(S_n(f))\). Later, W. Gehlen and W. Luh proved that there exists \(f\in H(\mathbb{D})\) such that for every closed \(C\subset\mathbb{C}\) with \(\partial\mathbb{D}\subset C\subset \mathbb{D}^c\) there exists a sequence \((n_k)\) of natural numbers such that \(Z(S_{n_k}(f))\cap \mathbb{D}^c=C\) [Arch. Math. 63, No. 1, 33–38 (1994; Zbl 0805.30002)]. The proof is constructive. In the same spirit of many results concerning universal series, the author shows in the paper under the review that this set of such functions is a \(G_\delta\) dense subset of \(H(\mathbb{D})\) (endowed with the topology of uniform convergence on compact subsets) using Baire’s theorem. Finally, the author combines this generic result with other classical universality properties to obtain new statements.