Can it be shown that this expression is decreasing?

Question

Suppose $x$ and $k$ are both odd integers such that $1 < k \leq x$. Is it true that the expression, \begin{align*} \prod_{i=1}^{x \left( \frac{k-1}{2} \right)} \left( \frac{x + 2i - 1}{x + 2i - 2} \right) \end{align*} is decreasing in $x$ for any such $k$? Graphing this expression would suggest so, but I have had difficulty showing it to be true.

Answer 1

For each term in the product:

$$\frac{x+2i-1}{x+2i-2} = 1+\frac{1}{x+2i-2}$$

Right side is decreasing in $x$, and finite product of decreasing functions is decreasing as well.