Solving an infinite summation involving exponential and factorial

Question

I'm trying to understand an equality that I found in this biology article.

$$\sum_{i=0}^\infty\frac{e^{-x}x^i(1-y)^i}{i!} = e^{-x\cdot y}$$

Can you help me proving this equation holds true?

Answer 1

The factor $e^{-x}$ does not change as $i$ runs through the list $0,1,2,3,\ldots$, so it can be pulled out, getting $$ e^{-x}\sum_{i=1}^\infty\cdots\cdots.$$

Then you have $\displaystyle\sum_{i=0}^\infty \frac{a^i}{i!} = e^a $, where $a=x(1-y)$.

Answer 2

There is a well known series expansion for $e^t$ for $t$ a real number.

Now $\sum\limits_{i=0}^{\infty}\dfrac{(x(1-y))^i}{i!}=e^{x(1-y)}$. Since $e^{-x}$ factor doesn't depend on $i$ we can pull it out. So we have that your = $e^{-x}e^{x-xy}$. Now use $e^{a+b}=e^ae^b$ to get your conclusion.