How to find pdf given moment generating function

[math-chick_41]

Given that the moment generating function of a random variable is
(e^t)/(2-e^t) is there a way I can go backwards and find the pdf, or could 2 different pdf's have the same mgf?

 

[balakrishnan_v]

Any bonafide moment generating function always maps to a unique distribution.
The random variable for the moment generating functi`on that you have given takes values at 1,2,3...infinity with probability 1/2,1/4,1/8...ie p(k)=1/2^k for k in natural numbers and zero otherwise.