Step 1:

The series is .

Expand the summation :

The series is alternating harmonic series.

Alternating series test :

If the alternating series   satisfies,

(i) for all .

(ii)   then the series is convergent.

Here and .

Thus, the first condition of the alternating test satisfied.

Step 2 :

Consider 

Find .

.

The second condition of the alternating test satisfied.

Hence the series is convergent.

Solution:

The series is is convergent.