The series is .

Ratio test :

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

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

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

Here and .

Find .

.

By the ratio test, the series is convergent when .

Radius of the convergence is half the width of the interval.

Radius of convergence is .

.

Check the interval of convergence at the end points.

For , .

The series is converges by -series test.

For , .

The series is conveges by alternating series test.

Therefore, interval of convergence is .

Radius of convergence is .

Interval of convergence is .