Answer
Diverges
Work Step by Step
We have: $a_k=\dfrac{1}{\sqrt k\sqrt {k+1}}$ and $b_k=\dfrac{1}{k}$
Now, we need to apply the limit comparison test.
$L=\lim\limits_{k \to \infty}\dfrac{a_k}{b_k}\\=\lim\limits_{k \to \infty}\dfrac{1/\sqrt k\sqrt {k+1}}{1/k}\\=\lim\limits_{k \to \infty}\dfrac{1}{\sqrt {1+\dfrac{1}{k}}}\\=\dfrac{\lim\limits_{k \to \infty} 1}{\lim\limits_{k \to \infty}\sqrt {1+\dfrac{1}{k}}}\\=\dfrac{1}{1+0}\\=1 \gt 0$
Therefore, the series diverges by the limit comparison test.
