Bound on expectation of function of standard normal, $\mathbb{E}[\exp(Z^a)]$

Question

I'm trying to find the maximum (or sup) of the value of $a$ such that $$\mathbb{E}[\exp(Z^a)]<+\infty$$ where $Z\sim \mathcal{N}(0,1)$. Obviously for $a=1$ the expectation is finite since it is the Log-normal. For $a=2$ it is $+\infty$ but I was no able to compute the expectation for $1<a<2$.

Answer 1

The problem is to determine the $a$ for which the integral $$I_a:=\int_0^{+\infty}e^{t^a-t^2}\mathrm dt$$ is convergent.

If $a\lt 2$, then $\lim_{t\to +\infty}\frac{|t^a-t^2|}{t^2}=1$, hence for $t$ large enough, $|t^a-t^2|\geqslant t^2/2$, and we conclude the convergence of $I_a$.