I am having a difficult time determining the support or range of a MGF for a given pmf. The specific questions states to find the MGF of f(x)=6/((x^2)(pi^2)) for x=1,2,3...
The result (which is the same answer on the prof's answer key) is
(6/pi^2) Sum (from x=1 to inf) (e^(tx))/x^2
The part that I do not understand is that on the answer key, it has t<=0. How is this figured out? I've looked online and through two different texts, and still am unsure.
The support of the $MGF$ is all values of $t$ such that the MGF exists, which in this case is equivalent to being finite. Your series is:
$$\sum_{x=1}^\infty \frac{e^{tx}}{x^2},$$
which diverges when $t>0$ since $\lim_{x\rightarrow\infty} e^{tx}/x^2\rightarrow=\infty\neq 0$, where 0 would a necessary condition for convergence. On the other hand, the series does converge for $t\leq 0$ by the comparision test to $\sum_{x=1}^\infty\frac{1}{x^2}<\infty$.
