help? comparision[integral]??

Question

Answer 1

(1)

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 .

Observe that .

Now compare the given series with the series .

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 divergent.

By comparison test, also diverges.

Solution :

is divergent.

Answer 2

(2)

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 n.

Observe that .

Now compare the given series with the series .

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 divergent.

By comparison test, also diverges.

Solution :

is divergent.

Answer 3

(3)

Step 1:

The series is .

Integral test :

Suppose is continuous , positive and decreasing on the interval then

(i) is convergent, then the is also convergent.

(ii) is divergent, then the is also divergent.

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

diverges, hence by integral test also diverges.

The series is divergent.

Solution :

The series is divergent.