Question

We have proved that the limit of $\sum_{k=0}^n r^2k^m / (1+r)^{k+1}$ when n approaches infinity is $\sum_{k=1}^m S(m,k)k!/r^{k-1}$ where S(m,k) is the second kind of stirling number.

Is there a simple asymptotic or approximate formula for the result $\sum_{k=1}^m S(m,k)k!/r^{k-1}$ with $m$ fixed and $r$ near $1$. ?

Answer 1

For the question to make sense, you have to specify what the asymptotics is with respect to. For example, which variables are fixed and which are going to infinity. If $r>0$ is fixed and $m\to\infty$ (and probably in some other cases too), you are better off analyzing your initial sum rather than the Stirling version. The largest term is around $k= m/\ln(1+r)$ and the terms near that have a Gaussian shape with standard deviation $m^{1/2}/\ln(1+r)$. Euler-Maclaurin summation for the main part plus crude bounds for the tails will give it to you.