Let $S_n=\prod_{k=3}^n (1-\tan^4\frac{\pi}{2^k})$. What is the value of $\lim_{n \to \infty} S_n$ ?
What I attempted:-
$\log S_n=\sum_{k=3}^n \log (1-\tan^4\frac{\pi}{2^k})$.
Since $\lim_{x \to 0} \frac{\tan x}{x}=1$, $\tan^4\frac{\pi}{2^k}\approx \left(\frac{\pi}{2^k}\right)^4$
Thus, $\log S_n=\sum_{k=3}^n \log (1-\frac{\pi^4}{2^{4k}})\approx \sum_{k=3}^n\left( -\frac{\pi^4}{2^{4k}}\right) $
Taking limit as $n \to \infty$, $\lim_{n \to \infty} \log S_n=\frac{-\pi^4}{3840}$.
Finally, $\lim_{n \to \infty}S_n=e^{\frac{-\pi^4}{3840}}\approx 1-\frac{\pi^4}{3840}$
Am I correct? Is there any other better method which yield the accurate limit? I was told that the exact limit should be one of the $4$ options:- $\frac{\pi^3}{4},\frac{\pi^3}{16}, \frac{\pi^3}{32},\frac{\pi^3}{256}$. I guess the third option is correct as it is giving almost the same value that I have obtained.
Best Answer
You obtained an approximation of the exact value. In order to find such exact vale, note that $$(1-\tan^4(\alpha/2))=(1+\tan^2(\alpha/2))(1-\tan^2(\alpha/2))= \frac{4}{\cos(\alpha)}\left(\frac{\tan(\alpha/2)}{\tan(\alpha)}\right)^2.$$ Hence, as $n$ goes to infinity, $$\prod_{k=3}^n \left(1-\tan^4(\pi/2^k)\right)=\frac{4^{n-2}}{\prod_{k=3}^n\cos(\pi/2^{k-1})}\cdot \left(\prod_{k=3}^n\frac{\tan(\pi/2^k)}{\tan(\pi/2^{k-1})}\right)^2\\=4^{n-2}\cdot 2^{n-2}\sin(\pi/2^{n-1})\cdot \left(\frac{\tan(\pi/2^n)}{\tan(\pi/2^{2})}\right)^2\to \frac{\pi^3}{32}$$ where we used the known fact that $$\prod\limits_{k=2}^{n}\cos\left(\frac{\pi }{2^{k}}\right)= \frac{1}{2^{n-1}\sin(\pi/2^n)}$$ (see for example How to evaluate $\lim\limits_{n\to \infty}\prod\limits_{r=2}^{n}\cos\left(\frac{\pi}{2^{r}}\right)$).