Answer
Diverges
Work Step by Step
Use a direct comparison test.
We can see that $0 \leq \dfrac{e}{x} \leq \dfrac{e^x}{x}$
Now, we will test $\dfrac{e}{x}$ for divergence.
So, $\int_{1}^{\infty} \dfrac{e dx}{x}=\lim\limits_{a \to \infty}\int_{1}^{a}\dfrac{e dx}{x} \\ =\lim\limits_{a \to \infty}[e \ln |x|]_{1}^{a} \\ \lim\limits_{a \to \infty}e \ln |a|-0\\=\infty$
Thus, the integral diverges by the direct comparison test.