For $p>0$, does $\int_1^\infty x^{-p/x}$ diverge?

Question

For $p>0$, does $\int_1^\infty x^{-p/x}$ diverge?
I've tried the root test, the comparison test, and the limit comparison test without success.
Any assistance would be appreciated.

Answer 1

Since $x=e^{\ln x}$, we are integrating $\exp(-(p\ln x)/x)$. Since $\frac{\ln x}{x}\to 0$ as $x\to\infty$, the integrand has limit $1$, and therefore in particular is after a while greater than $\frac{1}{2}$.

It follows that the improper integral diverges to $\infty$.

Answer 2

Hint1: Taylor series expansion. Hint2: Treat as complex function (asymptotic stability).

:)