$\"\"$

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 $\"\"$ .

$\"\"$

$\"\"$ .

$\"\"$

$\"\"$ .

Therefore, the series $\"\"$ is converges for $\"\"$ . $\"\"$

The series converges for $\"\"$ .