The series is .

Alternating Series Test:

If the alternating series  satisfies

(i)  .

(ii) .

then the series is convergent.

Verify condition (i) : 

Consider the related function .

Differentiate the function with respect to x . 

.

Since we are considering only positive , consider .

.

For , .

Verify condition (ii):

Apply L-hospital rule to find the limit.

LHospitals Rule :

 If the value of limit is indeterminate form of type  or , then

.

Series satisfies conditions of alternating series test.

Thus, the series is convergent.

The series is convergent.