I am trying to show that
$$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\prod_{j=0}^{i}\frac{kn-j-k}{kn-j}=\frac{k^{k+1}-(k-1)(k-1)^{k}}{(k+1)k^{k}}$$
for all $k\in\mathbb{N}$, $k\geq 4$.

I could verify the statement with Mathematica, but I could not find a self-standing proof. The product is telescopic, but I could only derive lower bounds instead of the exact value of the limit.