If E[X^r] = 6^r
where r = 1, 2, 3...
…can you please help me find the moment generating function and the p.m.f. of X?
From the definition of the moment generating function it should be clear that
$\displaystyle M_X(t) = 1 + 6t + \frac{6^2}{2!} t^2 + .... + \frac{6^n}{n!} t^n + .... = \sum_{k = 0}^{+ \infty} \frac{(6t)^k}{k!}$
and you should recognise what this function is. Then it should be clear that $\displaystyle M_X(t)$ is the mgf of a pmf that has all its mass concentrated at one point. That is, p(x) = 1 when x = c and the value of c is obvious.