$\lim_{n\to\infty}\prod_{r=1}^n\frac{4r^2}{4r^2+1}=\;?$

Question

$$\lim_{n\to\infty}\prod_{r=1}^n\frac{4r^2}{4r^2+1}$$

Please solve this question. I am unable to solve this.

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Answer 1

One may recall the infinite product of the $\sin$ function which yields $$ \frac{\sinh x}{x}=\prod\limits_{n=1}^\infty\left(1+\frac{x^2}{\pi^2 n^2}\right), \quad x\neq0, $$ then $$ \lim_{n\to\infty}\prod_{r=1}^n\frac{4r^2}{4r^2+1}=\left(\prod\limits_{r=1}^\infty\left(1+\frac{1}{4 r^2}\right)\right)^{-1}=\frac{\pi}{e^{\pi/2}-e^{-\pi/2}}. $$