Answer
Diverges
Work Step by Step
We have: $a_k=\tan (1/k)$ and $b_k=\dfrac{1}{k}$
Now, we need to apply the limit comparison test.
$L=\lim\limits_{k \to \infty}\dfrac{a_k}{b_k}\\=\lim\limits_{k \to \infty}\dfrac{\tan (1/k)}{1/k}$
Let us suppose that $p=\dfrac{1}{k} $ and when $k \to \infty$; then $p \to 0^{+}$ . So we will differentiate above to obtain :
$\lim\limits_{k \to \infty}\dfrac{\tan (1/k)}{1/k}=\lim\limits_{p \to 0^{+}}\dfrac{\sec^2 p}{1}=1$
Therefore, the series diverges by the limit comparison test.
