I have a series of the following form: \begin{align} \sum_{k=2}^\infty \left( 1 - e^{-ns^{k-1}} \right)^k \end{align} where $0<s<1$. I would like to compute an approximation of this series, for the regime where $n$ is large.
One natural thing to try would be to use the approximation $1-e^{-x} \approx x$, which is valid for small $x$. However, note that we are interested in large $n$, and the dominant terms of the series are those with small $k$, and thus $ns^{k-1}$ is relatively large for this approximation to be accurate. Also, even if it was an accurate approximation, I am not sure how to deal with the doubly exponential term that results from it.
Is there a better way to approximate this? I am particularly interested in a tight lower bound on this series, if it is practical to obtain one.
I would rather keep the exponential form and do binomial expansion for say $k\le4$ (you may want to try a different number) and ignore terms for $5<k$. Then the form would be
\begin{align} \sum_{k=2}^{\infty} \left(1-e^{-ns^{k-1}}\right)^k&\approx 1-e^{-ns}+1-2e^{-ns^2}+e^{-2ns^2}+1-3e^{-ns^3}+3e^{-2ns^3}-e^{-3ns^3} \end{align} I'm not sure how much better this will be than your current approximation but as far as $n$ is a big number, this might help.