Why is this not a valid moment generating function?

Question

Explain why there can be no random variable for which $M_x(t) = \frac{t}{1-t}$, where M is the moment generating function.

Attempt: I tried writing $\frac{t}{1-t}$ as the sum of an infinite series so $\sum t^n$ from $n=1$ to $\infty $. We know the formula for a moment generating function is $\sum e^{tx}f(x)$. So I compared the two and tried to show this leads to a density that does not integrate to 1, but I get that: $$ f(n) = \sum (\frac{t}{e^t})^n $$, which in general is convergent since $t<e^t$. How else would I show this?

Thanks!

Answer 1

There's no playing about with series needed, no worrying about integrals (well not directly). This is "think about what MGFs are" kind of question.

Consider the definition of an MGF -- $\quad M_X(t)=E(e^{tX})$.

Given that definition, what should $M_X(0)$ be?

Do you notice the problem?