<s>[INST] sharp bounds for $(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{k}{n})$ Is there any good bounds or estimation of $(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{k}{n})$ and $(1-\frac{1}{n})(1-\frac{2}{n})\cdots(1-\frac{k}{n})$, $1 < k < n$?
What I actually want is $\sum_{1\le k \le n}(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{k}{n})$, and $\sum_{1\le k \le n}(1-\frac{1}{n})(1-\frac{2}{n})\cdots(1-\frac{k}{n})$.
 [/INST] Classic products, whose results are $\left({\pm 1 \over n} \right)^k P(1\pm n,k)$ where $P$ is the Pochammer symbol.
And what you "really" want is:  
$$-\frac{2^{2 n+1} e^{-n} n^{n+1} \Gamma \left(\frac{1}{2} (2 n+3)\right) \Gamma (-2 n-1,-n)}{\sqrt{\pi }}+e^{-n} (-n)^{n+1} \Gamma
   (-n,-n)-1$$
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