If we look at this sequence

The first thing we should notice is that it is alternating from positive to negative. This means that we will have

.

The second thing we should notice is that the sequence is increasing in powers of 2.

Thus we will also have

.

Now we can combine these statements and write them in terms of a series.

We can now simplify this into 

.

To determine whether this alternating series converges or diverges, we must use the Alternating Series test, which states that for the series 

 and ,

where  for all n, if  and  is a decreasing sequence, then the series is convergent.

First, we must identify , which is . When we take the limit of  as n approaches infinity, we get

Notice that for the limit, the negative power terms go to zero, so we are left with something that does not equal zero.

Thus, the series is divergent because the test fails. 

To determine whether the series converges or diverges, we must use the Alternating Series test, which states that for 

 - and  where  for all n - to converge, 

 must equal zero and  must be a decreasing series.

For our series, 

because it behaves like 

.

The test fails because  so we do not need to check the second condition of the test.

The series is divergent.

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

1. 
2. {} is a decreasing sequence, or in other words 

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

1.        
2.        {} is a decreasing sequence, or in other words 

Solution:

1.

Our tests stop here. Since the limit of  goes to infinity, we can say that this function does not converge according to the alternating series test.

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

1.        
2.        {} is a decreasing sequence, or in other words 

Solution:

1.

This concludes our testing. The limit does not equal to 0, so we cannot say that this series converges according to the alternate series test.

Conclusive according to the alternating series test

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

1.        
2.        {} is a decreasing sequence, or in other words 

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

1.        
2.        {} is a decreasing sequence, or in other words 

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

1.        
2.        {} is a decreasing sequence, or in other words 

Solution:

1.

Conclude with tests. The limit of  goes to infinity and so we cannot use the alternating series test to reach a convergence/divergence conclusion.

This is an alternating series.

An alternating series can be identified because terms in the series will “alternate” between + and –, because of 

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

1.        
2.        {} is a decreasing sequence, or in other words 

Solution:

1.

2.

Since the 2 tests pass, this series is convergent.