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.

The series is .

.

.

Since is converges, .

.

For the series is converges.

For the series is diverges.

Therefore , the series  is converges for .

The series converges for .