# Thread: Converge and diverge of an improper integral

1. ## Converge and diverge of an improper integral

For which values of k , where k is a Real number does the improper integral:

integral( (ln(x)^k) / (x) ) from 1 to infinity

converge? For which values of k does it diverge? Justify your answer.

2. Originally Posted by Chris23
For which values of k , where k is a Real number does the improper integral:

integral( (ln(x)^k) / (x) ) from 1 to infinity

converge? For which values of k does it diverge? Justify your answer.
Consider that $\forall{k}\in[1,\infty)~\frac{\ln^k(x)}{x}\geqslant\frac{1}{x}$ which diverges. Now I want you to try out the other answers, the most important other number is when $k=-1$ Remember that you can integrate this as it is/.

3. ## RE: Converge or diverge of an improper integral

In the exercise I meant that k is a real number, can take any value from -infinity to +infinity. The improper integral has lower limit 1 and upper limit +infinity.
Do you know if there is any specific range in which the improper integral diverges and any other range that the improper definite integral converges?
And who I can find this? By taking limits?

Thanks a lot!

4. Originally Posted by Chris23
In the exercise I meant that k is a real number, can take any value from -infinity to +infinity. The improper integral has lower limit 1 and upper limit +infinity.
Do you know if there is any specific range in which the improper integral diverges and any other range that the improper definite integral converges?
And who I can find this? By taking limits?

Thanks a lot!
What I said was in response to it being a definite integral, I did not explicity state this but what I meant to say was that

$\int_1^{\infty}\frac{\ln^k(x)}{x}dx\geqslant\int_1 ^{\infty}\frac{dx}{x}=\infty~\forall{k\geqslant{0} }$