Step 1 :

Alternating Series Test :

If the alternating series satisfies

(i) 

(ii)

then the series is convergent.

Step 2 :

The series is .

Verify condition (i) :

Since the series is alternating, verify condition (i) and (ii) of the Alternating Series Test.

It is not obvious that the sequence given by is decreasing.

So consider the related function .

Differentiate the function with respect to x .

Since we are considering only positive x , consider .

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.

Thus, condition (i) verified.

Step 3 :

As , then .

Evaluate the limits.

.

Thus, condition (ii) is verified.

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

Solution :

The given series is convergent by the Alternating Series Test.