Question

Determine whether the series converges or diverges.

Answer 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 and the dominant part of the denominator is .

Now compare the given series with the series .

Observe that .

Because the denominators are equal and numerator is 1 greater in .

Answer 2

Contd.....

Step 2 :

The obtained series is .

Definition of p - series :

The p - series is convergent if and divergent if .

It is divergent because .

Therefore is divergent  by part (ii) of the Comparison Test.

Solution :

is divergent.