Simplification of a fractional factorial expresion

Question

I am stuck simplifying the following expression: $$\frac{\left(\frac{k-1}2\right)!}{\left(\frac k2\right)!}$$ I knew the result is $\frac k{2(k+1)}$ or $\frac k{k+1}$ but I am unable to prove it. In the expansion of these factorials, there is a half-integer difference between the numerator's and denominator's expansions.