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 1 and the dominant part of the denominator is .

Now compare the given series with the series .

Observe that . 

Step 2 :

The obtained series is .

Definition of p - series :

The p - series is convergent if and divergent if .

Here p = 1.

Hence the series is divergent.

Here and is divegent.

By comparison test, also diverges.

Step 3:

Solution :

is divergent.

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 1 and the dominant part of the denominator is .

Now compare the given series with the series .

Observe that .

Step 2 :

The obtained series is .

Definition of p - series :

The p - series is convergent if and divergent if .

Here p = 1.

Hence the series is divergent.

Here and is divegent.

By comparison test, also diverges.

Solution :

is divergent.

Step 1:

The series is .

Integral test :

Suppose is continuous , positive and decreasing on the interval then

(i) is convergent the is also convergent.

(i) is divergent the is also divergent.

The function is continuous , positive and decreasing on the interval .

The series is divergent.

Solution:

The series is divergent.