Let $\alpha$ be a strictly positive real and $X$ a discrete random variable admitting an exponential moment of order $\alpha$. Show that the random variable $e^{\alpha X}$ has finite expectation.
Let $\alpha$ be a strictly positive real and $X$ a discrete random variable admitting an exponential moment of order $\alpha$. Show that the random variable $e^{\alpha X}$ has finite expectation.