Determine the values of $p$ for which integral $\int_e^\infty \frac{\ln x}{x^p} dx$ converges

Question

Determine the values of $p$ for which integral $\int_e^\infty \frac{\ln x}{x^p} dx$ converges.

By integration by parts we get $\frac{x^{1-p}}{1-p}\left(\ln x - \frac{1}{1-p}\right)$, but I don't know how to go further since $\ln \infty \rightarrow \infty$.

Then it seems that maybe I should just reason about it directly, since $\int_1^\infty \frac{1}{x^p}$ diverges for $p \le 1$ and converges otherwise.

Please help refine my thinking by showing the best way to reason about this.

Answer 1

With $u:=\ln x$ your integral is $\int_1^\infty u\exp( -(p-1)u) \mathrm{d}u$, which converges iff $\Re (p)>1$.

Answer 2

As you noted it, $\log(x) \to \infty$, so the only way we can have a finite limit is if the other term in the product goes to $0$. When will $x^{1-p}\to 0$? Can you calculate the limit in that case? When will it be finite?