Let $S$ denote a random walk that satisfies Spitzer's condition $$ \frac{1}{n} \sum _{k=1}^n P (S_k > 0 ) \to \rho$$ for some $\rho \in (0,1)$. From the book Regular Variation (Bingham, Goldie, Teugels), the function $$ l(n) = \exp \left( \sum_{k=1}^\infty \frac{(1-\frac{1}{n})^k} {k} \Big( P(S_k > 0) - \rho \Big) \right) $$ is slowly varying. I am interested in learning more about the behavior of $l(n)^{-1}$ for $n$ large. In particular, I am asking myself whether the following statement can be made:

Let $\epsilon > 0$. Then, for $n$ large enough
$$  l(n)^{-1} \leq n^\epsilon.$$

I am thankful for any hints, references and ideas.