Let f be a holomorphic function on the unit disc. Abel’s Limit Theorem tells us that, if the Taylor series of f about 0 converges at a boundary point, then f has nontangential limit there and the two limits agree. No such conclusion may be drawn if we know merely that a given subsequence of the partial sums converges on the boundary (even if it converges on the whole unit circle). Nevertheless, as will be explained, there are strong connections between the behaviour of such sequences of partial sums on the unit circle and the boundary behaviour of f. The talk will also describe the subtle potential

theoretic tools used to establish these results, and applications to universal series. This is joint work with Myrto Manolaki.