Inverting a product

Question

Can anyone explain why $$\prod^{0}_{n=5}\frac{1}{f(n)}=f(1)f(2)f(3)f(4)$$ in other words is there some relationship or identity for dealing with inverses in products.

Answer 1

If you want to extend the general rule $$\prod_{k=l}^mf(k)\prod_{k=m+1}^nf(k)=\prod_{k=l}^nf(k),$$which is obviously true when $l\leqslant m< n$, to cases when $l$, $m$, and $n$ are in some other order, then your result follows (assuming that $f(k) \neq0$ in the range). However, not everybody adheres to this convention. Instead, a commonly adopted convention is simply "an empty product is unity": that is, $$\prod_{k=m}^nf(k)=1$$ whenever $n<m$ and not just when $n=m-1$.

Added: If you set $l=1$, $m=0$, and $n=1$ in the first identity, you get $$\prod_{k=1}^0f(k)\prod_{k=1}^1f(k)=\prod_{k=1}^1f(k),$$for all $f$. That is, $$\prod_{k=1}^0f(k)=1$$for all $f$. Using this result, substitute $l=1$, $m=4$, and $n=0$ in the first identity and you get the result in the question.