Alternating Series and Conditional Convergence   (continuing discussion)

 

 

In a Nut Shell:  An alternating series may converge but its series of absolute values

may diverge.  In such a case the alternating series is said to be conditionally convergent. 

 

 

 

The following series is an example of a conditionally convergent series.

                           ∞                                

                           ∑(-1)n (1/n)  

                        n = 1 

 

 

                                                                              ∞                                

Notice that the series of absolute values is           ∑ ( 1/n )  

                                                                            n = 1 

 

 is just the harmonic series, a positive term series, and is known to diverge. 

 

 

     Yet the alternating series

 

                           ∞                                

                           ∑(-1)n ( 1/n )  

                        n = 1 

 

     satisfies both conditions of the alternating series test namely that

 

     a.  an    ≥   an+1    ≤   0  for all n   and

 

     b.   lim  an   = 0 

        n → ∞

 

and thus the alternating series converges.  

 

 

 

                                                      ∞

So the original series                  ∑(-1)n (1/n)  

                                                  n = 1 

 

is conditionally convergent.

 

 

 

 

Click here for examples.

 



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