Answer
Diverges
Work Step by Step
Given: $\Sigma_{n =1}^{\infty} \cos \dfrac{1}{n}$
Since the function $\cos x$ is continuous at $0$, we have:
$\lim\limits_{n \to \infty} \dfrac{1}{n}=0$ and $\cos \lim\limits_{n \to \infty} \dfrac{1}{n}=\cos (0)=1\ne 0$
Hence, the series is divergent.
