I am assuming you mean discrete geometric distribution with x = 0,1,2, ....

on simplifying mgf = (1/2) ∑ [exp(t) / 2 ]^x where the summation is from 0 to infinity

mgf = (1/2) / { 1 -[exp(t) / 2 ] } using the formulae for infinite geometric series (with suitable restriction on t as you said)

note

expand the ∑ and use 1 + u + uČ + . . . = 1 / (1-u) if |u| < 1

here u = [exp(t) / 2 ]

try the following page for mgf of binomial distribution