How to show the integral $\int_e^\infty \left(\frac{e}{t}\right)^t dt$ converges? Let $$I=\int_e^\infty \left(\frac{e}{t}\right)^t dt$$
How to show it converges?
I tried to find some inequality to compare with.

By replacing $t$ with $e u$ we have:
$$ I = e \int_{1}^{+\infty}\frac{du}{u^{eu}}\leq e\int_{1}^{+\infty}\frac{du}{u^e}=\frac{e}{e-1}.$$
A more tight inequality is given by:
$$ \forall u\geq 1,\qquad \frac{1}{u^{eu}}\leq \exp\left(e(1-u)\right) $$
from which it follows that $\color{red}{I\leq 1}$.
