I want to calculate $$\lim_{N \to \infty} \prod_{j=1}^{N} \frac{2j}{N + j + 1} \,.$$ Here is my logic:
- For the range of the product operator ($1 \leq j \leq N $) the inequality $2j < N + j + 1$ is always true. Therefore $\frac{2j}{N + j + 1} < 1 $, and so an upper boundary can be estabilished using the largest term of the product: $$\lim_{N \to \infty} \prod_{j=1}^{N} \frac{2j}{N + j + 1} \leq \lim_{N \to \infty}\frac{2N}{2N + 1} = 1 \,.$$
However, when I use WolframAlpha I get $\infty$ for an answer.
I have seen other questions that challenge the reliability of WolframAlpha, so I am tempted to dismiss it.
My questions:
- Is my logic correct or is WolframAlpha correct? (and why?)
- If the limit is indeed finite, does it go to zero or does it simply coneverge to some value smaller than 1? How can I show it?
Thank you very much.