Hint for $\lim_{n\rightarrow\infty} \sqrt[n]{\prod_{i=1}^n\frac{1}{\cos\frac{1}{i}}}$. How to calculate the following limit:
$$\lim_{n\rightarrow\infty}\sqrt[n]{\prod_{i=1}^n\frac{1}{\cos\frac{1}{i}}}$$
thanks.

Apply $\ln$ to get
$$\tag 1\frac{1}{n}\left (\sum_{k=1}^{n}\ln [1/\cos (1/k)].\right)$$
Now $\ln [1/\cos (1/k)] \to 0$ as $k\to \infty.$ Therefore $(1),$ which are the Cesaro means of these terms, also converge to $0.$ Hence the limit of the original expression is $1.$
