Universal Taylor series on unbounded open sets

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Universal Taylor series where the universal approximation is valid on the boundary of the open unit disc can not be smooth or bounded. If the universal approximation is not requested to be valid on the boundary, then it is well known that the universal function can be bounded and smooth. We extend this result to unbounded open sets Ω, such that {∞} ∪ (C – Ω¯) is connected.