Evaluation of convergence of series $$\sum^{\infty}_{k=1}\frac{5\sqrt{k}}{6k^2\sqrt{k}-2k+7}$$ using camparasion or limit Camparasion Test
What i try
$$\sum^{\infty}_{k=1}\frac{5\sqrt{k}}{6k^2\sqrt{k}-2k+7}\approx\sum^{\infty}_{k=1}\frac{5\sqrt{k}}{6k^2\sqrt{k}}=\frac{5}{7}\sum^{\infty}_{k=1}\frac{1}{k^2}.$$
Which is converge using $p$ series test.
So our original series is converges.
But i have a problem i have just approximate it.can anyone explain me a fair way to do it. Thanks
It is enough to show that
$$\frac{5\sqrt{k}}{6k^2\sqrt{k}-2k+7}<\frac1{k^2}.$$