Module 9 :  Infinite Series, Tests of Convergence,  Absolute and Conditional Convergence, Taylor and Maclaurin Series
Lecture 26 :  Absolute convergence [Section 26.1]
26.1.3 Theorem (The ratio Test):
 

Let be a series of positive terms such that
.
Then the following hold:

(i)

If , then the series is convergent.

(ii)
 If or , then the series is divergent.
(iii)
 If ,the series may converge or diverge.
  Proof
(i)
 For , select such that , and choose such that
         .
Then
         .
Thus
         .
Hence, for
         
Since is a convergent geometric series, as ,by comparison test, is convergent.
 
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