Step 1:

The series is .

The Ratio Test:

Let  be a series with nonzero terms.

1) converges absolutely if .

2) diverges if and .

3) The Ratio Test is inconclusive if .

Step 2:

Consider .

Apply Ratio Test.

This series is convergent because is less than .

Solution:

The series is convergent.

Step 1 :

The series is .

The Comparison Test :

Suppose that and are series with positive terms.

(i) If is convergent and for all n , then is also convergent.

(ii) If is divergent and for all n, then is also divergent.

The dominant part of the numerator is  and the dominant part of the denominator is .

Observe that .

Now the obtained series is .

Step 2:

The series is .

Test of divergence : If doesnot exist or , then the series is divergent. \ \

By the test of divergence, the series is divergent.

Here and is divergent.

By comparison test, is also divergent. \ \

Solution:

The series is divergent.