TITLE: Computation of limit: $\lim\limits_{n \to \infty} \frac{1}{n}\left ( \cos1+\cos(\frac{1}{2})+\cos(\frac{1}{3}) +...+\cos (\frac{1}{n}) \right )$
QUESTION [6 upvotes]: Help me please to compute this limit:
$\lim\limits_{n \to \infty} \frac{1}{n}\left ( \cos1+\cos(\frac{1}{2})+\cos(\frac{1}{3}) +...+\cos (\frac{1}{n}) \right )$
Thank you.

REPLY [5 votes]: We may nicely squeeze it 
$$\lim\limits_{n \to \infty}\sqrt[n]{\cos (1)\times\cos (1/2)\times\cdots\times\cos(1/n)}\le\lim\limits_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n}\cos\left(\frac{1}{k}\right)\le\lim\limits_{n \to \infty}\left(\frac{1}{n}\times n\cos(1/n)\right)$$
$$1\le\lim\limits_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n}\cos\left(\frac{1}{k}\right)\le1$$
where on the left side I combined AM-GM and Cauchy-d'Alembert criterion.