Determine whether the series converges absolutely or conditionally, or diverges.

To determine the convergence or divergence of the series , we may apply Alternating Series Test.

In Alternating Series Test, the series is convergent if:

1) is monotone and decreasing sequence.

2)

3)

For the series , we have:

which is a positive, continuous, and decreasing sequence from

Note: As " " increases, the increases then decreases.

...

To determine the convergence or divergence of the series , we may apply Alternating Series Test.

In Alternating Series Test, the series is convergent if:

1) is monotone and decreasing sequence.

2)

3)

For the series , we have:

which is a positive, continuous, and decreasing sequence from

Note: As " " increases, the increases then decreases.

Then, we set-up the limit as :

By alternating series test criteria, the series  converges.

The series  has positive and negative elements. Thus, we must verify if the series converges absolutely or conditionally. Recall:

a) Absolute Convergence:    is absolutely convergent if   is convergent.  

b) Conditional Convergence:   is conditionally convergent if  is divergent and  is convergent.  

We evaluate the as :

Applying integral test for convergence, we evaluate the series as:

Apply u-substitution: then .

                       

                       

Plug-in on the indefinite integral  , we get:

Applying definite integral formula: .

Then, the limit becomes:

                                   

                                    )|

                                   

implies the series   diverges.

Conclusion:  

The series is conditionally convergent since as   is divergent and  as is convergent.