I am stuck in computing this series (i.e, finding a closed-form formula):
$$ \sum_{i=0}^k \binom{k}{i} \frac{2r^{i+1}(1-r)^{k-i+1}p^{k-i}v^i s^k}{(1-r)p^{k-i}s^i + r v^i s^{k-i}}, $$ where $r$, $p$, $v$, $s$ are constants in the range of $(0,1)$. Any help/hint/comment would be greatly appreciated.