Random algorithm termination

Question

Suppose I have an algorithm that works as follows when invoked: it calls itself recursively with probability $0 < p < 1$ and terminates with probability $1-p$. Does this algorithm terminate? On the one hand you can show that $\lim_{n \to \infty}p^n = 0$, so it must terminate at some point before infinity. On the other hand it also seems reasonable to say that it is possible for the algorithm to call itself repeatedly indefinitely. What is the correct interpretation?