Answer
Diverges
Work Step by Step
We have: $a_k=\dfrac{1}{2k-\sqrt k}$ and $b_k=\dfrac{1}{k}$
Now, we need to apply the limit comparison test.
$L=\lim\limits_{k \to \infty}\dfrac{a_k}{b_k}\\=\lim\limits_{k \to \infty}\dfrac{\dfrac{1}{2k-\sqrt k}}{1/k}\\=\lim\limits_{k \to \infty}\dfrac{k}{2k-\sqrt k} \\=\lim\limits_{k \to \infty}\dfrac{1}{2-(1/k^{1/2})^k}\\=\dfrac{1}{2}$
Therefore, the series diverges by the limit comparison test.
