The integral Test :

If  is positive, continuous and decreasing for  and  then  and either both converge or both diverge.

The series is .

(a)

For .

Substitute  in .

.

Therefore, the sum of the series is  is diverges for .

The series is .

(b)

For .

Substitute  in .

Therefore, the sum of the series is  is diverges for .

The series is .

(c)

For .

Find the value for  is converge where .

Since , consider .

.

.

The series is  is converges if .

Therefore, the sum of the series is  is converges if .

(a) The sum of the series is  is diverges for .

(b) The sum of the series is  is diverges for .

(c) The sum of the series is  is converges if .