I have a quite basic question about Poisson CDF. I have the following function $P(x;\mu)$ referring to the Poisson CDF.

Hadley and Whitin (1963) defined the Poisson density as

$p(x;\mu)= \frac{\mu^x}{x!}e^{-\mu}$

and complementary cumulative distribution function as

$P(x;\mu)=\sum\_{j=x}^{\infty}\frac{\mu^j}{j!}e^{-\mu}$.

**To calculate $P(x;\mu)$, can I use the following:**

**$P(x;\mu)=1-\sum\_{j=0}^{x-1}\frac{\mu^j}{j!}e^{-\mu}$**

**knowing that**

**$\sum\_{j=0}^{\infty}\frac{\mu^j}{j!}e^{-\mu}=1$?**

I have seen several applications of CDF calculation online and they made me confused. Can you help to verify?