Let $m_X(t)$ be the moment generating function of random variable $X$. Prove $m_X(t)=\sum_{k=0}^\infty E(X^k)\frac{t^k}{k!}$

Question

Let $m_X(t)$ be the moment generating function of random variable $X$. Prove $m_X(t)=\sum_{k=0}^\infty E(X^k)\frac{t^k}{k!}$

So I have:

$$ \begin{split} m_X(t) &= \mathbb{E}\left[ e^{tX} \right]\\ &= \mathbb{E}\left[ \sum_{k=0}^\infty \frac{(tX)^k}{k!} \right] \\ &=\sum_{k=0}^\infty \mathbb{E}\left[ X^k \right] \\ &=\sum_{k=0}^\infty \frac{t^k}{k!} \mathbb{E}\left[ X^k \right] \end{split} $$

Answer 1

To make the question precise I will assume that $Ee^{tX} <\infty$ for all $t$. Since $e^{|tX|} \leq e^{tX}+e^{-tX}$ we get $Ee^{|tX|} <\infty $ and so $E\sum_n \frac {|t^{n} |X^{n}} {n!} <\infty$. We can now invoke Fubini/Tonelli Theorem to justify the interchage of the sum and the expecation.