In a recent answer I gave a combinatorial interpretation for the sum $\sum_{n=1} \binom{2n}{n}\frac{4^{-n}}{n+1}=1$: namely, that it corresponded to the probability of all outcomes adding to $1$. A few commentators objected that I was taking for granted that such a game had no chance of lasting forever. That was a valid point, though one I felt comfortable leaving out of an intuitive explanation...
However, that left me wondering about the following question. What are some good examples of terminating random walks with finite 'escape' probability? (That is, a random what such that the stopping probability does not go to one as the number of steps goes to infinity.) As a secondary question: What are some notable summations implied by such walks?
A simple example is a biased $\pm1$ random walk with $p\gt1/2$ probability of a $+1$ step, starting from $1$, stopped when it first hits $0$. Probability of "escaping" to infinity is $2-1/p$.
"Hence", for every $p$ in $(0,1)$, $$\sum_{n=0}^{\infty}\frac1{n+1}{2n\choose n}p^n(1-p)^n=\left\{\begin{array}{ccc}\frac1p&\text{if}&p\gt\frac12\\\frac1{1-p}&\text{if}&p\leqslant\frac12\end{array}\right.$$ (Which is the extension to every $p$ of the summation you recalled for $p=1/2$ at the beginning of your post.)