Theorem 4.4. Let and . Then (converges) if and only if both (converge).
Proof.
Proof of Theorem 4.4 is in the book.
Complex Analysis for Mathematics and Engineering
![[Graphics:Images/ComplexSequenceSeriesMod_gr_134.gif]](../Images/ComplexSequenceSeriesMod_gr_134.gif){kind=small}
and ![[Graphics:Images/ComplexSequenceSeriesMod_gr_135.gif]](../Images/ComplexSequenceSeriesMod_gr_135.gif){kind=small}
. Then ![[Graphics:Images/ComplexSequenceSeriesMod_gr_136.gif]](../Images/ComplexSequenceSeriesMod_gr_136.gif){kind=small}
(converges)![[Graphics:Images/ComplexSequenceSeriesMod_gr_137.gif]](../Images/ComplexSequenceSeriesMod_gr_137.gif){kind=small}
(converge). and . Then (converges) (converge).