I have to find, for all value of constant , the interval of converence of .

How do I do? By using the Ratio test I get

Now I will use the rule of Hopital to make this fraction 'simpler'.

According to the Ratio test, it states that this series is convergent if , that is . Divergent if , that is . If , it has not result. If we test it by inserting the value of on this series, it would be divergent according to Wolfram Alpha. I don't understand this part because this problem clearly states the value of a should be greater than 0, so should also be convergent.

The other hand if we see it converges if where , so I am not really sure I could say it would be . If it's true, the interval of convergent would be like as .

If I did it wrong, please tell me what to do.