![[Graphics:Images/ComplexGeometricSeriesMod_gr_61.gif]](../Images/ComplexGeometricSeriesMod_gr_61.gif){kind=small}
, the
series ![[Graphics:Images/ComplexGeometricSeriesMod_gr_62.gif]](../Images/ComplexGeometricSeriesMod_gr_62.gif){kind=small}
converges
to ![[Graphics:Images/ComplexGeometricSeriesMod_gr_63.gif]](../Images/ComplexGeometricSeriesMod_gr_63.gif){kind=small}
. That
is, if ![[Graphics:Images/ComplexGeometricSeriesMod_gr_64.gif]](../Images/ComplexGeometricSeriesMod_gr_64.gif){kind=small}
then (4-11)
![[Graphics:Images/ComplexGeometricSeriesMod_gr_65.gif]](../Images/ComplexGeometricSeriesMod_gr_65.gif){kind=small}
. If
![[Graphics:Images/ComplexGeometricSeriesMod_gr_66.gif]](../Images/ComplexGeometricSeriesMod_gr_66.gif){kind=small}
, the
series diverges. Theorem 4.12 (Geometric
Series).
If , the
series converges
to . That
is, if then
(4-11) .
If , the
series diverges.
Proof.
Proof of Theorem 4.12 is in the book.
Complex Analysis for Mathematics and Engineering