Answer
Diverges
Work Step by Step
We have: $a_k=\dfrac{\cot h k }{k}$ and $b_k=\dfrac{1}{k}$
Now, we need to apply the limit comparison test.
$L=\lim\limits_{k \to \infty}\dfrac{a_k}{b_k}\\=\lim\limits_{k \to \infty}\dfrac{\dfrac{\cot h k }{k}}{1\ k}\\=\lim\limits_{k \to \infty} \cot h k \\=1$
We see that $0 \lt \lim\limits_{k \to \infty}\dfrac{a_k}{b_k} \lt \infty$
and the series $\Sigma b_k$ is a divergent harmonic series.Therefore, the given series diverges by the limit comparison test.
