What is the value of $$\prod_{i=2}^n \frac{2i^2-i-1}{2i^2+i-1}$$ when $n$ approaches infinity?
2026-04-02 00:29:21.1775089761
What is the limit of $\prod\limits_{i=2}^n\frac{2i^2-i-1}{2i^2+i-1}$ when $n$ approaches infinity?
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Hint. Note that $$\frac{2i^2-i-1}{2i^2+i-1}=\frac{(2i+1)(i-1)}{(2i-1)(i+1)}.$$ Therefore \begin{align*}\prod_{i=2}^n\frac{2i^2-i-1}{2i^2+i-1}&=\frac{\prod_{i=2}^n(2i+1)}{\prod_{i=2}^n(2i-1)}\cdot \frac{\prod_{i=2}^n(i-1)}{\prod_{i=2}^n(i+1)} \\&=\frac{\prod_{i=2}^n(2i+1)}{\prod_{i=1}^{n-1}(2i+1)}\cdot \frac{\prod_{i=1}^{n-1}i}{\prod_{i=3}^{n+1}i} =\frac{2n+1}{3}\cdot\frac{1\cdot 2}{n(n+1)}. \end{align*} Can you take it from here?