Answer
Diverges
Work Step by Step
Given $$\cos \frac{1}{2}+\cos \frac{1}{3}+\cos \frac{1}{4}+\cdots=\sum_{n=1}^{\infty}\cos\frac{1}{n+1}$$
Since
\begin{align*}
\lim_{n\to\infty} \cos\frac{1}{n+1}&= \cos 0\\
&=1\neq0
\end{align*}
Then the series diverges.
