Answer
Converges
Work Step by Step
We have: $a_k=\dfrac{2}{k^2-10}$ and $b_k=\dfrac{1}{k^2}$
Now, we need to apply the limit comparison test.
$L=\lim\limits_{k \to \infty}\dfrac{a_k}{b_k}\\=\lim\limits_{k \to \infty}\dfrac{\dfrac{2}{k^2-10}}{1\ k^2}\\=\lim\limits_{k \to \infty} \dfrac{2k^2}{k^2-10}\\=\lim\limits_{k \to \infty} \dfrac{2k^2}{1-10/k^2}\\=2$
We see that $0 \lt \lim\limits_{k \to \infty}\dfrac{a_k}{b_k} \lt \infty$
and the series $\Sigma b_k$ convergent p-series with $p=2$.Therefore, the given series converges by the limit comparison test.
