I'm studying stochastic processes and in the chapter of simple random walk there's an example saying let $\mu_b$ the mean number of visit of the state $b$, starting at $S_0=0$ and without returning in $0$. The book said: $\mu_b=\sum_n P(S_1\cdot...\cdot S_{n-1}\neq 0, S_n=b)$.
To understand this fact I'm thinking: let $X=$number of visit of the state $b$, starting at $S_0=0$ and without revisiting $0$. $X=\sum_n I_\{S_1\cdot...\cdot S_{n-1}\neq 0, S_n=b\}$.
My problem is that i need some conditions to switch expectation with the symble of series.