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I am studying the asymptotic behavior of the sum $$S_n = \sum_{p=1}^{n} \frac{p!}{n!}$$ By expanding this sum, I have observed that it follows a pattern resembling the series $$S_n = 1 + \frac{1}{n} + \frac{1}{n^2} + \frac{2}{n^3} + \frac{5}{n^4} + \dots + \frac{B_k}{n^k} + o\left(\frac{1}{n^k}\right)$$ where the coefficients are very likely to be the Bell numbers.

However, I am uncertain whether this connection is genuine or merely coincidental. Any insights or pointers to relevant literature would be greatly appreciated.

RobPratt
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Roanoke Yehia
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1 Answers1

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This is true up until the $(n-1)$-th term, and also your indexing is off by $1$. It's clearer what's going on if we reindex the sum starting from its largest term; so, we write

$$\begin{align*} S_n &= \sum_{k=0}^{n-1} \frac{(n-k)!}{n!} \\ &= \sum_{k=0}^{n-1} \frac{1}{n(n-1) \dots (n-k+1)} \\ &= \sum_{k=0}^{n-1} \frac{1}{n^k \prod_{i=0}^{k-1} \left( 1 - \frac{i}{n} \right)} \\ &= 1 + \sum_{k=1}^{n-1} \frac{1}{n^k} \sum_{m=k-1}^{\infty} \left\{ m \atop k-1 \right\} \frac{1}{n^{m-k+1}} \\ &= 1 + \sum_{m=0}^{\infty} \frac{1}{n^{m+1}} \sum_{k=1}^{n-1} \left\{ m \atop k-1 \right\} \\ &= \boxed{ 1 + \sum_{m=0}^{\infty} \frac{B_{m, n-1}}{n^{m+1}} } \end{align*}$$

where $B_{m, n-1}$ denotes the number of partitions of a set of size $m$ into at most $n-1$ non-empty subsets and in the fourth line we use the ordinary generating function of the Stirling set / partition numbers.

Qiaochu Yuan
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