Question:
Determine for which values of $\mu$ a birth-and-death process with $\mu_n=\mu$ and $\lambda_n=2+\cos (n \pi)$ admits steady state probabilities $\left\{p_n\right\}_{n \geq 0}$.
My thought:
This may meaning that: $$C_n=\frac{\lambda_{n-1} \lambda_{n-2} \cdots \lambda_0}{\mu_n \mu_{n-1} \cdots \mu_1}$$ where $$ \sum^{\infty}_{n=1} C_n \neq \infty $$
However, I think it is too simple. I am curious what things I missed?