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 $1$. ?