So i have derived this expression and would like to simplify it (i.e find an expression purely in terms of J and v).
$$ J\bigg[1+\sum_{n=1}^{J-1}\prod_{k=1}^{n}\frac{k(1-v)(J-k)}{(k+1)(J-k-1 + vk)}\bigg]$$
So far we can get
$$ J\bigg[1+\sum_{n=1}^{J-1}\frac{(1-v)^n}{n+1}\prod_{k=1}^{n}\frac{(J-k)}{(J-k-1 + vk)}\bigg]$$ I know that $0 \leq v \leq 1$ and that $J$ is an integer $\geq 1$
I've tried shoving it into Mathematica and have not managed to obtain a simpler form for the full expression. Any suggestions or is this the best i can get?