I am looking for an approximation for the following sum for large $n$
$$\sum_{k=1}^n(-1)^{k+1}\binom nk\frac{1}{1-(1-p)^k}$$
where $p\in(0,1)$.
It would help, if there were an expression for this sum. But I can't find one.
For those interested: the origin of this sum is probabilistic. It is about the waiting time until n distinguishable dice show 6. Each round all the dice are thrown, those that show a six are singled out. Clearly the waiting time is $T=\max(W_1,W_2,\ldots,W_n)$, where $W_i\sim Geo(p=1/6)$
Since the geometric distribution behaves nicely with regards to the minimum i used the following formula
$$\mathbb E[\max(W_1,\ldots,W_n)]=n\mathbb E[W_1]+\sum_{i=1}^n(-1)^i\binom ni\mathbb E[\min(W_1,W_2,\ldots,W_i)]$$
The formula stems from a inclusion exclusion principle. The formula is also known as mini-max-equation for expectations.
However, i want to look at this quantity for large $n$ and need a starting point. I tried the Stirling-Formula, but a closed expression would be nicer.
Maybe someone can put it into mathematica?
