Prove the series converges $\sum_{k=2}^{\infty} \sqrt{\frac{k^2+1}{k^3-3}}$

Question

Prove $\sum_{k=2}^{\infty} \frac{\sqrt{k^2+1}}{k^3-3}$ converges.

I was trying to find an easy comparison for an integral test, but not much luck.

Answer 1

$\sqrt {k^{2}+1} \leq k+1$ and $k^{3}-3 \geq \frac 1 2 k^{3}$ for all $k >1$. Can you complete the argument?

Answer 2

It is easy to see that $\frac{k^2+1}{k^3-3} \ge \frac{1}{k^2}$ for $k \ge 2$, hence

$\sqrt{\frac{k^2+1}{k^3-3}} \ge \frac{1}{k}$ for $k \ge 2$.

Consequence: the series is divergent.