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.

So 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) verified.

Apply L hospital rule :

As , then .

.

Thus, condition (ii) is verified.

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

The series is convergent.