I am looking for an equivalent (or something weaker) of the following integral:

$u_n=\int_{1}^{+\infty}e^{-t}\log^n(t)\,\mathrm dt$

when $n\rightarrow\infty$. I have tried some recurrence relations that leads to $u_n^\frac{1}{n}=o(n^\epsilon)$ for any $\epsilon>0$, but I'm sure it is possible to get something sharper. Any hint is welcome! Thank you.

I am looking for an equivalent (or something weaker) of the following integral:

$$u_n=\int_{1}^{+\infty}e^{-t}\log^n(t)\,\mathrm dt$$

when $n\rightarrow\infty$. I have tried some recurrence relations that leads to $u_n^{\frac{1}{n}}=o(n^{\epsilon})$ for any $\epsilon>0$, but I'm sure it is possible to get something sharper. Any hint is welcome! Thank you.