Let $X$ be uniformly distributed on the interval $(0,1)$. Then $X$ has the moment generating function $M_x(s)=\frac{e^s-1}{s}$. I am attempting to find all moments of $X$, and I would normally expand the MGF, but I don't know if this can be done when $M_x(s)$ is not defined on an open interval containing $0$.
Can I still expand to to $M_x(s)=\sum_{k=0}^{\infty}\frac{s^k}{k!}E[X^k] $? Or is there some other method to use?
I would go back to the basics. Note that $$ EX^k=\int_0^1 x^k\times1\,dx=\frac{1}{k+1}. $$