Result of product with n=0

Question

It's a simple question but I couldn't find informations about it and I'm starting to learn product sequences.

I noticed (using WolframAlpha) that:

$\prod_{i=x}^{0}{f(i)}|_{x > 0} = 1$

Why is that? Shouldn't a sequence of zero products equals zero?

Ex : $\prod_{i=x}^{0}{f(i)}|_{x > 0} = 0$?

Answer 1

We want $\prod_{j=1}^n a_j = a_n\cdot\prod_{j=1}^{n-1}a_j$ to hold as generally as possible. The requires $a_1=\prod_{j=1}^1a_j=a_1\prod_{j=1}^0 a_j$, so the convention that the empty product equals $1$ is adequate. Compare also with $0!=1$ because there is exactly one way to arrange no objects, or with $x^0=1$.

Answer 2

An empty product equals $1$. This is convenient for when the notation appears, for example, in the denominator.

https://en.wikipedia.org/wiki/Multiplication#Capital_Pi_notation