Let $K \leq 3$ an integer. Prove that $$\underset{i=0}{\overset{K}{\sum}}\frac{\left(-1\right)^{i}}{i!}\frac{\left(K-i-1\right)^{2}}{\left(K-i\right)!}=0.$$
It is true if you reverse the inequality. If , it holds. Here is a rough outline for a proof:
For :
Edit: The crux of the proof involves trying to make each term resemble the binomial formula for for some power .