TITLE: Find the value of $\lim_{n\to \infty } \,\cos (1) \cos \left(\frac{1}{2}\right) \cos \left(\frac{1}{4}\right)\cdots \cos \left(\frac{1}{2^n}\right)$
QUESTION [7 upvotes]: $$\lim_{n\to \infty } \,\cos (1) \cos \left(\frac{1}{2}\right) \cos \left(\frac{1}{4}\right)\cdots \cos \left(\frac{1}{2^n}\right)$$
How would you evaluate this limit? Is it just equivalent to $$\prod_{n=0}^\infty {\cos \left( \frac{1}{2^{n}} \right)}$$

REPLY [12 votes]: Multiply first by $2^{n+1}\sin(1/2^n)$ and collapse it.