Answer
Convergent with limit 1
Work Step by Step
We are given the sequence:
$a_n=\cos\left(\dfrac{2}{n}\right)$.
As $n$ increases, $\dfrac{2}{n}$ decreases, so $\cos\left(\dfrac{2}{n}\right)$ increases; therefore $a_n$ is a monotonic sequence.
On the other hand, $\cos\left(\dfrac{2}{n}\right)$ is a bounded sequence because $\cos x$ is a bounded sequence.
So, according to the Monotonic Sequence Theorem, the sequence $a_n$ is convergent.
Calculate the limit:
$\displaystyle{\lim_{n \to \infty}} \cos\left(\dfrac{2}{n}\right)=\cos \left(\displaystyle{\lim_{n \to \infty}}\left(\dfrac{2}{n}\right)\right)=\cos 0=1$
Therefore the sequence converges and its limit is 1.
