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.

For $t > 2e$, you have 
$$\left(\frac et\right)^t \le \left(\frac 12 \right)^t $$
and of course 
$$\int_{2e} ^\infty \left(\frac 12 \right)^t dt $$ converges
