<s>[INST] Evaluate $\lim\limits_{n\to\infty}\prod\limits_{k=2}^{n}\frac{k^2+k-2}{k^2+k}$ I can't find the product of a sequence.
We have
$$\frac{(2+2)(2-1)}{2(2+1)}\frac{(3+2)(3-1)}{3(3+1)}...\frac{(k+2)(k-1)}{k(k+1)}$$
I am stuck with $$P=\frac{2(n+2)}{n^2(n-1)}$$ but that isn't correct. Can the squeeze theorem be used?
 [/INST] $$\prod_{k=2}^{K}\frac{k^2+k-2}{k^2+k}=\prod_{k=2}^{K}\frac{k-1}{k}\prod_{k=2}^{K}\frac{k+2}{k+1}=\frac{1}{K}\cdot\frac{K+2}{3}$$
hence the limit equals $\color{red}{\displaystyle\frac{1}{3}}$.
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