Alternating Series Test :

If the alternating series  satisfies

(i)  ,

(ii) , then the series is convergent.

The series is .

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.

Consider the related function .

Differentiate the function with respect to x .

If the function is decreasing, then .

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) is verified.

Find .

As , then .

Evaluate the limits.

.

Thus, condition (ii) is verified.

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

The series is convergent.