Rooted quivers are quivers that do not contain
A∞≡⋯→•→• as a subquiver. The existence of flat covers and cotorsion envelopes
for representations of these quivers have been studied by Enochs et al. The main goal of this paper is to prove that flat covers and cotorsion envelopes exist for
representations of A∞. We first characterize finitely generated projective representations of A∞. We also see that there are no
projective covers for representations of A∞, which adds more
interest to the problem of the existence of flat covers.