The known proofs for universal Taylor series do not determine a specific universal
Taylor series. In the present paper, we isolate a specific universal Taylor series by
modifying the proof in [30]. Thus we determine all Taylor coefficients of a specific
universal Taylor series on the disc or on a polygonal domain. Furthermore in non
simply connected domains, when universal Taylor series exist, we can construct a
sequence of specific rational functions converging to a universal function, provided
the boundary is good enough. The solution uses an infinite denumerable procedure
and a finite number of steps is not sufficient. However we solve a Runge's type
problem in a finite number of steps.

Let Ω be a simply connected proper subdomain of the complex plane and z0 be a point in Ω. It is known that there are holomorphic functions f on Ω for which the partial sums (Sn(f,z0)) of the Taylor series about z0 have universal approximation properties outside Ω. In this paper we investigate what can be said for the sequence (βnSn(f,z0)) when (βn) is a sequence of complex numbers. We also study a related analogue of a classical theorem of Seleznev concerning the case where the radius of convergence of the universal power series is zero.

It is known that, for any simply connected proper subdomain Omega of the complex plane and any point zeta in Omega, there are holomorphic functions on Omega that have "universal" Taylor series expansions about zeta; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in C\Omega that have connected complement. This note shows that this phenomenon can break down for non-simply connected domains Omega, even when C\Omega is compact. This answers a question of Melas and disproves a conjecture of Müller, Vlachou and Yavrian.