Answer
Diverges
Work Step by Step
We have: $a_k=\sqrt {\dfrac{k}{k^3+1}}$ and $b_k=\dfrac{1}{k}$
Now, we need to apply the limit comparison test.
$L=\lim\limits_{k \to \infty}\dfrac{a_k}{b_k}\\=\lim\limits_{k \to \infty}\dfrac{\sqrt {\dfrac{k}{k^3+1}}}{1/k}\\=\lim\limits_{k \to \infty} \sqrt {\dfrac{k^3}{k^3+1}}\\=1$
Therefore, the series diverges by the limit comparison test.
