Consider a birth-death process with constant parameters $\lambda_n = 4$ and $\mu_n = 5$ with $n = 0,1,2,\dots$. Find the stationary probability (as a function of n)
My attempt so far is using the balance equations $\lambda_n\pi_n = \mu_{n+1}\pi_{n+1}$ we have the equations $4\pi_0 = 5\pi_1, ..., 4\pi_n = 5\pi_{n+1}$ would yield $\pi_1 = \frac{4}{5}\pi_0, \pi_2 = \frac{4}{5}\pi_1 = (\frac{4}{5})^2\pi_0, \dots,\pi_n = (\frac{4}{5})^n\pi_0$
and solving for $\pi_0 = (1+\frac{4}{5} + (\frac{4}{5})^2 + \dots + (\frac{4}{5})^n)^{-1} = (\frac{1}{1-\frac{4}{5}})^{-1} = 5$. Then putting it all together my answer of $\pi_n = (\frac{4}{5})^n\cdot5$ was incorrect. I hope someone will be able to lead me in the correct direction
The detailed balance equations $$ \pi_{n-1}\lambda_{n-1} = \pi_n\mu_n,\ n=1,2,\ldots $$ yield the recurrence $$\pi_n = \prod_{i=1}^n \frac{\lambda_{i-1}}{\mu_i}\pi_0,$$ and from $\sum_{n=0}^\infty \pi_n=1$ it follows that \begin{align} \pi_0 &= \left(\sum_{n=0}^\infty\prod_{i=1}^n \frac{\lambda_{i-1}}{\mu_i}\right)^{-1}\\ \pi_n &= \prod_{i=1}^n \frac{\lambda_{i-1}}{\mu_i}\left(\sum_{n=0}^\infty\prod_{i=1}^n \frac{\lambda_{i-1}}{\mu_i}\right)^{-1}. \end{align} Substituting $\lambda_n=4$ and $\mu_n=5$ for all $n$, we have $$ \pi_0 =\left(\sum_{n=0}^\infty\prod_{i=1}^n \frac45\right)^{-1} = \left(\sum_{n=0}^\infty\left(\frac45\right)^n\right)^{-1}=\frac15 $$ and $$ \pi_n = \left(\prod_{i=1}^n \frac45\right)\cdot\frac15 = \frac15\left(\frac45\right)^n. $$
Your reasoning looks correct, but you neglected to take the factor of $-1$ when computing $\pi_0$.