# Summation with factorial

I want to understand how this step is performed. Can you tell me that how this value of Po is obtained from the first equation.!

$$P_o=\frac{1}{\sum_{k=0}^\infty\left(\frac{\alpha}{u}\right)^k\frac{1}{k!}}$$

$$P_o=e^{-\frac{\alpha}{u}}$$

• It's rather unreadable. Could you type it? Commented Mar 31, 2015 at 15:20
• Is it better now.? Commented Mar 31, 2015 at 15:21
• It appears false to me. Can't see all that well, but it appears that where there is multiplication by $k!$, it would need to instead be division. Commented Mar 31, 2015 at 15:21
• yes sorry, it is 1/k! Commented Mar 31, 2015 at 15:25
• I've edited it and now it is 1/k!. Commented Mar 31, 2015 at 15:30

$$P_o=\frac{1}{\sum_{k=0}^\infty\left(\frac{\alpha}{u}\right)^k.\frac{1}{k!}}$$
Substitute $\frac{\alpha}{u}=x$.
The denominator is $$f(x)=1+x+\frac{x^2}{2!}+\dots$$
Notice that this is the Taylor expansion for $e^x$, and this yields the answer. $$f(x)=e^x$$ $$P_0= \frac{1}{f(x)} = e^{-\frac{\alpha}{u}}$$