Determination of a universal series

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.