(a)

and are series with positive terms.

Suppose that and converges.

From the definition of the limit of convergence there exist such that When .

Here

So for .

Which means .

If converges then so does .

Hence by the comparison test is converges.

(b)

(i)

The series is .

Consider .

Consider and it is converges by -series test.

Find .

.

If and converges then the series is converges.

Therefore, is converges.

(ii) The series is .

Consider .

Consider and it is converges by geometric series test.

Find .

.

If and converges then the series is converges.

Therefore, is converges.

(a) is converges.

(b)

(i) is converges.

(ii) is converges.