The series is  .

Here .

Find the convergence of the series by Ratio test.

Ratio Test :

(i) If , then the series is  is absolutely convergent.

(ii) If  or , then the series is  is divergent.

(iii) If , then the ratio test is inconclusive.

Find .

For ,.

The series is divergent for .

For ,

For , it is a rational function with numerator having smaller degree than the denominator.

Therefore series converges for  all .

series converges for  all .