Given the following sum:

$S(n) = \sum_{i=1}^{n} \frac{1}{1-\operatorname{H}(p)}$

where $H$ is the binary entropy function defined as:

$\operatorname{H}(p) = -p\log p - (1-p)\log (1-p) $.

Let $f(n) = \frac{n}{S(n)}$.

Assume $p$ is very small, is it possible to approximate the $S(n)$ and $f(n)$ defined above with simpler closed formulae e.g. a polynomial of $p$ without loosing much precision?