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 .

If , then

.

Thus, condition (i) is not verified.

Find .

As , then .

The function is oscillates in between and .

Thus, condition (ii) is not verified.

Thus the given series is divergent .

The series is divergent.