The series is .

Alternating Series Test :

If the alternating series  satisfies

(i)  ,

(ii) ,

Then the series is convergent.

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 .

Apply derivative on each side with respect to .

.

, for .

Therefore, the series is decreasing.

Since the series is decreasing .

Find .

.

.

Since the series is decreasing and , is converges.

The series is convergent.

The series is convergent.