Can $cos(u)$ be a moment generating function?

Question

I was asked in the probability course:

Can a moment generating function of a random variable $Y$ be $\cos(u), -\infty <u<\infty$?

I proved that the first property of moment generating functions is satisfied, as in:

$1=M_Y(0) = \cos(0) = 1$

But I couldn't carry on logically. What is missing so I could prove that indeed it can be? (My intuition says it can)

Answer 1

Try looking at $E[Y^2]$ (which you know has to be $\geqslant 0$).