How find this limits $\lim_{n\to\infty}(\frac{2}{1^4}+1)(\frac{2}{2^4}+1)(\frac{2}{3^4}+1)\cdots(\frac{2}{n^4}+1)$

Question

How to Find this limit $$\lim_{n\to\infty}\left(\dfrac{2}{1^4}+1\right)\left(\dfrac{2}{2^4}+1\right)\left(\dfrac{2}{3^4}+1\right)\cdots\left(\dfrac{2}{n^4}+1\right)$$

see 1

I remeber in sack have solve this follow problem: But I can't find it $$\prod_{n=1}^{\infty}\left(\dfrac{1}{n^4}+1\right)$$ May be we use same methods can solve it.

Answer 1

(Note to the reader: If you find this answer rather terse, then, first, you are right and, second, I suggest that you read the comment thread.)

$$2\pi^2x^2\cdot\prod_{n=1}^{\infty}\left(1+\frac{x^4}{n^4}\right)=\cosh(\pi\sqrt2x)-\cos(\pi\sqrt2x)$$