show this $$\prod_{n=1}^{N}\left(1+\dfrac{1}{n^2}\right)<4\tag{1}$$
I have know this can find the value $$\prod_{n=1}^{+\infty}\left(1+\dfrac{1}{n^2}\right)=\dfrac{\sinh{\pi}}{\pi}=3.67<4$$Prove that $\prod_{k=2}^{+\infty} (1+1/k^2) = \sinh(\pi)/(2 \pi)$.
But I think the problem $(1)$ have Simple proof?That is, we avoid calculating the result to give proof with inequality? Thanks
Note that $$\prod^N_{n=2} \left(1-\frac1{n^2}\right) = \prod^N_{n=2} \frac{(n-1)(n+1)}{n^2} = \frac12\cdot\frac{N+1}{N} > \frac12,$$ and that
$$\prod^N_{n=2}\left(1+\frac1{n^2}\right)\prod^N_{n=2}\left(1-\frac1{n^2}\right) =\prod^N_{n=2}\left(1-\frac1{n^4}\right)< 1.$$ So $$\prod^N_{n=2}\left(1+\frac1{n^2}\right) < 2,$$ and we are done.