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January 26th, 2018, 03:55 AM  #1 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 103 Thanks: 2  absolute convergence of complex series.
Let $z_{n}$ be a complex series such that $\frac{z_{n+1}}{z_{n}}$ tends to $x$ as n tends to infinity where x<1. Show $z_{n}$ converges absolutely.

January 26th, 2018, 06:31 AM  #2 
Senior Member Joined: Oct 2009 Posts: 771 Thanks: 278  
January 26th, 2018, 08:00 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,636 Thanks: 2621 Math Focus: Mainly analysis and algebra 
$z_{n+1} < \alpha z_n$ makes no sense in the complex plane.

January 26th, 2018, 09:01 AM  #4 
Senior Member Joined: Oct 2009 Posts: 771 Thanks: 278 
Not canonically at least. But I'm sure the OP can fix this small issue. 
January 26th, 2018, 12:29 PM  #5 
Global Moderator Joined: May 2007 Posts: 6,730 Thanks: 689  

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absolute, complex, convergence, series 
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