Any ideas how to upper bound $A$, $B$, $C$ and/or $D$:
$$ B = \sum_{i=1}^n \frac{s_i}{i + {s_i}} $$
$$ C = \sum_{i=1}^n \frac{1}{i + \sqrt{s_i}} $$
$$ D = \sum_{i=1}^n \frac{1}{i + {s_i}} $$
as a function of $S = \sum_{i=1}^n s_i$, $P = \sum_{i=1}^n \sqrt{s_i}$ and/or $\sum_{i=1}^n 1/i = O(\log n)$ (and possibly some other things)?
Update: The assumption is that $s_i \geq 1, \forall i$.