Module 9 :  Infinite Series, Tests of Convergence,  Absolute and Conditional Convergence, Taylor and Maclaurin Series
Lecture 26 :  Conditional convergence [Section 26.2]
  Then by ratio test, the series is absolutely convergent.
The relation between convergence and absolute convergence of a series is described in the next theorem.
26.2.4

Theorem:

  If a series is absolutely convergent, then it is also convergent.
 
26.2.5 Examples:
 

Let

Note that, is not a geometric series. However, is a geometric series with common-ratio . Hence, is absolutely convergent, and thus is itself convergent.

Finally, we give a test which helps us to analyze convergence of an alternating series.

26.2.6 Theorem (Alternating series test):
 

Let be an alternating series such that
(i)
(ii) Then is convergent.

 
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