# Simplify a equation containing factorial, summation, and fraction

$$\large P_0=\frac{1}{\left[\sum_{i=0}^{M-1} \frac{1}{i!}\left(\frac{\lambda}{\mu}\right)^i\right]+\frac{1}{M!}\left(\frac{\lambda}{\mu}\right)^M\frac{M\mu}{M\mu-\lambda}}$$
The sum can be summed in terms of incomplete gamma-function $\Gamma_x(s) = \int_{x}^\infty y^{s-1} \mathrm{e}^{-y} \mathrm{d} y$: $$\sum_{i=0}^{M-1} \frac{1}{i!} \frac{\lambda^i}{\mu^i} = \exp(\frac{\lambda}{\mu}) \frac{\Gamma_{\frac{\lambda}{\mu}}\left(M \right)}{\Gamma(M)}$$