For what values of p does the series from n1 to infinity n3p

For what value(s) of p does the series from n=1 to +infinity ((n^(3p))+1)/(sqrt(n+9)) converge?

Answer is \'no values\'.

Solution

Consider following the series:

Here, both are series with positive terms.

Use the limit comparison test.

Take .

Here, is diverges, then is diverges.

Suppose that, is converges form some , then the summation should be converges. Thus, gives either both series are converges or first series should be diverges to and second series diverges to , which are contradiction.

Thus, our supposing is wrong.

Therefore, is not converges for any values of . That is, there is no such values.