Let $f_n(x)=\prod_{k=0}^n \frac{1}{x+k}$.
I need to show for every $x \in \mathbb{R}, x>0$:
$\sum\limits_{n=0}^{\infty} f_n(x) = e\sum\limits_{n=0}^{\infty} \frac{(-1)^n}{(x+n)n!}$
What I have noticed: they have the same evaluation for $x=1$: they both go to $e-1$. Also, the left one is easy to evaluate for all natural numbers.
I'm quite sure there is some smart move with Taylor expansion, even though it doesnt't seem.