I'm stuck with an exercise here. So there's a random walk on the integers with probability 0.5 of going left or right, i.e. $P[X_i=1]=P[X_i=-1]=1/2$ and $S_n=X_1+...+X_n$. for a real number $c$, set
$M_n=e^{cS_n}(\frac{2}{e^c+e^{-c}})$ and show that $M_n$ is a martingale.
What does that mean? Do I have to show that we have a martingale measure, i.e. some risk-neutral probabilities that sum up to 1? Would that be $\sum_{n \in \mathbb{Z}} M_n=1$? I don't see how to apply the definition on this, if I'm even taking the right definition here. Can someone please explain me what I exactly have to do?