# $\prod_{n=3 }^{+\infty} \cos\frac{\pi}{n}$is convergent or not?

$$\prod_{n=3 }^{+\infty} \cos\frac{\pi}{n}$$is convergent or not?

Note that $$\lim_{n \to \infty} \ln{\cos\frac{\pi}{n}}=0,$$I try to turn it into $$\sum_{n=3}^{+\infty}\ln{\cos\frac{\pi}{n}}$$but have no idea how to do next.

• Hint: Compute $\lim_{x\rightarrow 0}\frac{\ln\left(\cos(\pi x)\right)}{x^2}$. Apr 27, 2022 at 3:57

Alternative approach:

For $$r \in \{3,4,5,\cdots\},$$ let $$a_r$$ denote $$\displaystyle \prod_{n=3}^r \left[\cos\left(\frac{\pi}{n}\right)\right].$$

So, the question is whether the infinite sequence $$\langle a_r\rangle$$ is a convergent sequence.

The sequence is strictly decreasing, and bounded below by $$0$$. Therefore, the sequence must be convergent.

let $$a_n=\ln\cos\frac{\pi}{n}$$, $$a_n<0$$.

hence $$\ln\cos\frac{\pi}{n}=\ln\left(1+\cos\frac{\pi}{n}-1\right)\\ \sim\cos\frac{\pi}{n}-1\sim-\frac{\pi^2}{2n^2}\quad(n\rightarrow \infty )$$

since we have known $$\sum_{n=1}^{\infty} -\frac{\pi^2}{2n^2}$$is convergent. $$\sum_{n=3}^{+\infty}\ln{\cos\frac{\pi}{n}}$$is convergent.