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  .

LHospitals Rule :

 If the value of limit is indeterminate form of type  or , then

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 ,

.

Divergence test:

If does not exist or if , then the series is divergent.

.

The series is divergent by divergence test.

The series  is convergent by alternating series test.

For ,

is divergent.

Therefore, interval of convergence is .

Radius of convergence is .

Interval of convergence is .