The series is .

Ratio test :

Let be a series with non zero terms.

1. converges absolutely if .

2. diverges if or .

3. The ratio test is inconclusive if .

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 convergence by alternating series test.

Therefore, interval of convergence is .

Radius of convergence is .

Interval of convergence is .