I want to prove $ \int_0^\infty \frac{e^{-x}}{x} dx = \infty $ How can I prove this integral diverges?
$$ \int_0^\infty \frac{e^{-x}}{x} dx = \infty $$

$s>0$
$$ \int_0^\infty e^{-sx} dx = \frac{1}{s} $$
$$ \int _1^\infty \int_0^\infty e^{-sx} dx ds = \int _1^\infty \frac{1}{s}ds $$
$$ \int_0^\infty  \int _1^\infty e^{-sx}  ds dx = \int _1^\infty \frac{1}{s}ds $$
$$ \int_0^\infty   (\frac{e^{-sx}}{-x}) |_{s=1}^\infty   dx = \int_1^\infty \frac{1}{s}ds $$
$$ \int_0^\infty  \frac{e^{-x}}{x} dx  = \ln\infty -\ln 1 =  \infty$$
