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Strictly the proof of
Theorem 7.6
says that either
-
the An are all infinite, in which case the radius of convergence is zero, or
-
the An are all finite, and converge to zero, in which case the radius of
convergence is infinite, or else
-
the An converge to something finite and non-zero, which is the
reciprocal of the radius of convergence.
You should read the proof of 7.12 carefully, and check for yourself that one or other
of the first two cases hold for
An if and only if the same case holds for
Bn. Thus, 7.12 holds for all possible values of
R.