Consider the sum $$S(n,c) = \sum\_{i=1}^{n-1}\dfrac{i}{ci+(n-i)}$$ where $0\le c\le 1$. 

When $c=0$, $S(n,c)$ grows asymptotically as $n\log n$.

When $c=1$, $S(n,c)$ grows asymptotically as $n$.

What about when $0<c<1$? Can we calculate $S(n,c)$ exactly? What about asymptotics? Can we find upper/lower bounds?