Hi, is it possible to give an explicit formula for the function G(s) defined for positive s as


$G(s) := \lim\_{N\to\infty} \sum\_{k=1}^N \frac{1}{k}{N\choose k} \left(\frac{s}{N}\right)^k \left(1-\frac{s}{N}\right)^{N-k}$.


Wolfram Mathematica says the sum for finite $N$ is some hypergeometric function, the limit of it as $N\to\infty$ is not able to express explicitely.


Thanks, Ian



Looking for an explicit formula for a limit of a binomial-like expression
Unless I've made some horrible miscalculation your limit is the same as
$$\lim\_{N\to \infty}\left(1-\frac{s}{N}\right)^N \int\_{0}^{\frac{s}{N-s}}\frac{(1+x)^N-1}{x} \ dx$$
which is equal to
$$e^{-s} \lim\_{N\to \infty} \int\_{0}^{\frac{Ns}{N-s}}\frac{\left(1+\frac{y}{N}\right)^N-1}{y} \ dy=e^{-s}\int\_{0}^s \frac{e^x-1}{x} \ dx.$$
And so you get the final answer
$$e^{-s}\left(Ei(s)-\gamma -\log s\right)$$
where $Ei(x)$ is the [Exponential integral](http://en.wikipedia.org/wiki/Exponential_integral) and can not be "simplified" further.


