Extra Example 2.  Show that   [Graphics:Images/ComplexPowerSeriesMod_gr_202.gif]   for all  [Graphics:Images/ComplexPowerSeriesMod_gr_203.gif]. 

Explore Extra Solution 2.

Use the fact that  [Graphics:../Images/ComplexPowerSeriesMod_gr_204.gif]  and  [Graphics:../Images/ComplexPowerSeriesMod_gr_205.gif].  

[Graphics:../Images/ComplexPowerSeriesMod_gr_206.gif]

[Graphics:../Images/ComplexPowerSeriesMod_gr_207.gif]

 

 

 

Or sum the infinite series directly.

[Graphics:../Images/ComplexPowerSeriesMod_gr_208.gif]

[Graphics:../Images/ComplexPowerSeriesMod_gr_209.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell