(a) The series is .

Find the convergence of the series by Ratio test.

Ratio Test :

(i) If , then the series is  is absolutely convergent.

(ii) If  or , then the series is  is divergent.

(iii) If , then the ratio test is inconclusive.

Here .

.

Find .

.

Thus the test is inconclusive.

(b)

The series is .

Here .

.

Find the convergence of the series by Ratio test.

Find .

.

Thus by the ratio test series is absolutely convergent.

(c)

The series is .

Here .

.

Find the convergence of the series by Ratio test.

Find .

Thus by the ratio test series is absolutely divergent.

(d)

The series is .

Here .

.

Find the convergence of the series by Ratio test.

Find .

Thus the test is inconclusive.

Therefore test is inconclusive for (a) and (d).

Ratio test is inconclusive for (a) and (d).