$\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}).$

Question

Please, can you help me to find the limit of the example

where n goes to infinity ?

Answer 1

Hint: write cos(x/2) = sinx/(2sin(x/2)), and cos(x/4) = sin(x/2)/(2sin(x/4)), etc...

Answer 2

The answer is $\frac{\sin x}{x}$.

See to this question's #1 answer: $\prod_{k=3}^{\infty} 1 - \tan( \pi/2^k)^4$

Answer 3

Just for your information (since you received good answers for the limit), the product from $i=1$ to $i=n$ of $\cos \left(2^{-i} x\right)$ has a closed form which is $$2^{-n} \sin (x) \csc \left(2^{-n} x\right)$$