![[Graphics:Images/ComplexPowerSeriesMod_gr_21.gif]](../Images/ComplexPowerSeriesMod_gr_21.gif){kind=small}
. Then the set of points z for which the series converges
is one of the following:(i) The single point
![[Graphics:Images/ComplexPowerSeriesMod_gr_22.gif]](../Images/ComplexPowerSeriesMod_gr_22.gif){kind=small}
.
(ii) The disk
![[Graphics:Images/ComplexPowerSeriesMod_gr_23.gif]](../Images/ComplexPowerSeriesMod_gr_23.gif){kind=small}
, along
with part (either none, or some or all) of the
circle ![[Graphics:Images/ComplexPowerSeriesMod_gr_24.gif]](../Images/ComplexPowerSeriesMod_gr_24.gif){kind=small}
. (iii) The entire complex plane.
Theorem
4.15. Suppose
. Then the set of points z for which the series converges
is one of the following:
(i) The single
point .
(ii) The
disk , along
with part (either none, or some or all) of the
circle .
(iii) The entire complex
plane.
Proof.
Proof of Theorem 4.15 is in the book.