Answer
Diverges
Work Step by Step
We have: $a_k=k \sin \dfrac{1}{k}$ and $b_k=\dfrac{1}{k}$
Now, we need to apply the limit comparison test.
$L=\lim\limits_{k \to \infty}\dfrac{a_k}{b_k}\\=\lim\limits_{k \to \infty}\dfrac{k \sin (1/k)}{1/k}\\=\lim\limits_{k \to \infty} (k) \times \lim\limits_{k \to \infty} \dfrac{\sin (1/k)}{1/k}\\=\infty \times 1 \\=\infty$
Therefore, the series diverges by the limit comparison test.
