Alternating Series Test

If the alternating series

satisfies

then the series is convergent.

EXAMPLE 1:  The alternating harmonic series  

Satisfies 

So the series is convergent by the Alternating Series Test.

EXAMPLE 2:  The series

is alternating, but  

so condition (ii) is not satisfied. Instead, we look at the limit of the th term of the series: 

This limit does not exist, so the series diverges by the Test for Divergence.

EXAMPLE 3:  Test the series 

for convergence or divergence.

The given series is alternating so we try to verify conditions (i) and (ii) of the Alternating Series Test. Unlike the situation in Example 1, it is not obvious that the sequence given by   is decreasing. However, if we consider the related function  ,  we find that  

Since we are considering only positive we see  , that is, . Thus   is decreasing on the interval   . This means that  and therefore    , when  . (The inequality  can be verified directly but all that really matters is that the sequence  is eventually decreasing.) 

Condition (ii) is readily verified:

Thus the given series is convergent by the Alternating Series Test.