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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 $1$. ?

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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 more simple asymptotic formula for the result $\sum_{k=1}^m S(m,k)k!/r^{k-1}$ ?

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We have proved that the limit of $r^2 * k^m \sum_{k=0}^n r^2k^m / (1+r)^(k+1)$ is $\sum_{k=1}^m S(m,k)k!/r^(k-1)$ 1+r)^{k+1}$when n approaches infinity is$\sum_{k=1}^m S(m,k)k!/r^{k-1}$where S(m,k) is the second kind of stirling number. Is there a more simple asymptotic formula for the result$\sum_{k=1}^m S(m,k)k!/r^{k-1}\$ ?

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