Module 9 :  Infinite Series, Tests of Convergence,  Absolute and Conditional Convergence, Taylor and Maclaurin Series
Lecture 26 :  Conditional convergence [Section 26.2]
(iii)
 Consider alternating series
 

      
Let
      
Then
      
Since
     ,
is a monotonically decreasing function. Thus
    
Further
   
Hence, by alternating series test, the above series is convergent.

26.2.8 Note:
(i)

The alternating series test not only gives the convergence of the series, in fact, if

 

,
then
.
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