As the title says, I need to show that $f(t)=\prod_{k=1}^\infty \cos\left(\frac{2\pi t}{3^k}\right)$ does not vanish at infinity, i.e. $f\not \in C_0(\mathbb{R})$.
I tried looking at the series
$$ \sum_{k=1}^\infty \left(1-\cos\left(\frac{2\pi t}{3^k}\right)\right)$$
in order to see that $f$ is actually well defined (i.e. the infinite product converges) and hoping that something could be read off the series (or the theorems on convergence of infinite products), but I'm having trouble seeing that said series is convergent.
I'd appreciate a hint.
Thanks in advance!