Vasco, my understanding is that on section 9.2 Penrose is talking about any

. If

the series will converge.

In section 9.3 he is talking about an

where

.

In 9.2 (pg 160) he says: "Does this mean that, for convergence of the Fourier series, we necessarily require the unit circle to lie within the annulus of convergence?

This would indeed be the case (...) "

And then in section 9.3 he is talking about a (any?) series that

will converge because

. My understanding is that we are not worried anymore about what happens inside/outside the unit circle, we are just worried about the unit circle itself ...

Does this make sense for you?

Rodrigo