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.

(a).

.

Therefore from the ratio test if ,or , then the series

is  is divergent.

Here .

Since , the series is divergent.

(b).

.

Therefore from the ratio test if , then the series is  is absolutely convergent.

Here .

Since , the series is absolutely convergent. 

(c).

.

Therefore from the ratio test if , then the ratio test is inconclusive.

This means that the ratio test does not tell us anything about the convergence or divergence

of the series .

The series may be converging or diverging, use some other test to find the convergence or divergence of the series.

(a). The series is divergent.

(b). The series is absolutely convergent.

(c). The ratio test is inconclusive.