Does this infinite product converge? And how to express it neatly?

Question

First of all, does this product have a "nicer" functional form--i.e., analogous to how you can write geometric sums in a nice closed expression:

$$(x-0)(x-1)(x-2)...(x-n)$$

Secondly, does this converge to something as n goes to infinity?

Answer 1

If $x$ is a non-negative integer $m$, the product is equal to $0$ for all $n\ge m$ and therefore converges to $0$. In all other cases it fails to converge.

To see this, let $a_n=x^{\underline{n+1}}=\prod_{k=0}^n(x-k)$. (Another notation is $(x)_{n+1}$, but I don’t care for it.) There is an $m$ such that $(x-m)<-2$, and clearly $|a_{n+1}|>2|a_n|$ for $n\ge m$, so

$$|a_n|\ge 2^{n-m}|a_m|$$

for $n\ge m$.