The function 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 .

Consider and .

Find .

.

The series is converges .

.

Check the end points.

For , .

The series is converges by -series test.

For , .

The series is converges by alternating series test.

The interval of converges for is .

Consider .

Apply derivative on each side with respect to .

.

Consider and .

Apply ratio test:

Find .

.

The series is converges when .

.

Check the end points.

For , .

The series is diverges by -series test.

For , .

The series is converges by alternating series test.

The interval of converges for is .

Consider .

Apply derivative on each side with respect to .

.

Consider and .

Apply ratio test:

Find .

.

The series is converges when .

.

Check the end points.

For , .

The series is diverges by divergence test.

For ,

The series is oscillates between  to .

Therefore, series is diverges.

The series is converges by alternating series test.

also diverges. \ \

The interval of converges for is .

The interval of converges for is .

The interval of converges for is .

The interval of converges for is .