Limit of the expectation of the sum

Question

Show that for $g(t)= E \left\{\sum_{n=3}^{\infty}\frac{(iut)^{n}}{n!}\right\}$ that $\lim_{t \to 0} \frac{|g(t)|}{t} =0$.

I think I should bound it and then use LDCT, but I'm having trouble doing that.

Thanks.