Here I have a moment generating function, $M_X(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{te^u} e^{- \frac{u^2}{2}} du$
I want to prove it does not exist for any t. But it seems to be impossible to directly solve this integrand.
I know that as $u \to \infty$, $te^u - \frac{u^2}{2} \to \infty$ because exponential term goes much faster than square. However, I don't know how to formally prove this, any help or hint is welcome