Finding the density of $X$

Question

Let $X$ be a random variable with $ E(X^m) = (m+1)! (2^m), \ m=1,2,3, ...$

Find the density of $X$.

I tried finding the nth derivative of the moment generating function to solve this question but somehow I can't find the answer.

Answer 1

https://en.wikipedia.org/wiki/Moment-generating_function This Wikipedia article describes the expansion of the moment generation function in terms of the moments (which you have).