Let $P\left ( X_{i}=1 \right )=p$ and $P(X_{i}=-1)=1-p$ where $p \in \left ( 0,1 \right )$ and $p\neq \frac{1}{2}$.
Let $S_{n}=S_{0}+X_{1}+\cdot \cdot \cdot X_{n}$. Define $M_{n}=\left ( \frac{1-p}{p} \right )^{S_{n}}$
Recall: if $h\left ( x \right )=\sum_{y}p\left ( x,y \right )h\left ( y \right )$ then $h\left ( X_{n} \right )$ is a martingale.
The author begins with
$\left ( p\left ( \frac{1-p}{p} \right )^{S_{n}+1} +\left ( 1-p \right )\left ( \frac{1-p}{p} \right )^{S_{n}-1}\right )$
which I can understand that the author takes into account the fact that the Markov chain could either move forward or backward and multiply that by its associated probability.
However, I do not understand why the summation vanishes.
Any help is appreciated.