power series and analytic functions

When I am doing homework I meet the following problem: suppose a function f is analytic in the open unit disc and continuous in the closed unit disc. So it could be represented by power series of in the open unit disc. So, does the continuity of f imply that its power series converge not only in the open unit disc but also in the close unit disc?

flag	asked Nov 14 at 7:22 Sun 13●1

The answer is no, but in special cases (e.g. if the Taylor series is absolutely summable) then one does get convergence at each point of the closed unit disc. However, it is true that you can get a sequence of polynomials which will converge uniformly on the (closed) unit disc to $f$ -- they just won't necessarily be the truncations of the Taylor expansion. – Yemon Choi Nov 14 at 7:28

Thanks, but could you give me a counter example? – Sun Nov 14 at 7:35

I think something like $a_n = n^{-1} \exp(in\log n)$ is supposed to be an example, but I am going from memory here. Katznelson's book on harmonic analysis may have more references – Yemon Choi Nov 14 at 8:35

Thank you very much! – Sun Nov 14 at 13:50

I voted to close. This is not the place to ask homework questions. – Alexandre Eremenko Nov 15 at 3:32

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