How do I sum 1/6( e^t + e^2t + .... + e^6t)?

Question

I have a question about a sum when calculating moment generating function. The question is : "Find the moment generating function for each of these two random variables. (i) $X$ = outcome a die toss, $p(x) = P[X= x] = \frac{1}{6}$ for $x = 1,2,3,4,5,6$. I don't understand how the sum in the answer $e^t \cdot \frac{1}{6} + e^{2t} \cdot \frac{1}{6} +...+ e^{6t}\cdot \frac{1}{6} = \frac{1}{6}e^t\frac{e^{6t}-1}{e^t-1}$. Why doesn't it work to have this as the sum of a finite geometric series with the common ratio being $e^t$?

Answer 1

I think it doesn't work with finite geometric series because $e$ is not rational number, but I am not sure