The series is .

Root test :

Let be a series.

1. converges absolutely if .

2. diverges if or .

3. The root test is inconclusive if .

Here .

Find .

.

Interval of convergence is .

The series is convergence for all values of and the radius of the convergence is half the width of the interval.

Radius of convergence is .

Radius of convergence is .

Interval of convergence is .