Ratio test :

The ratio test for absolute convergence of is :

Let

1.If , then the series converges absolutely.

2. If , then the series diverges.

3.If , then additional investigation required.

The series is .

Consider .

The value of  .

Substitute in .

Apply L hospital rule.

If then .

.

When the series is converges .

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

.

.

Therefore, the series is converges in the interval .

The radius of the convergence is and the series is converges in the interval .