Test for convergence/divergence of $\sum_{k=1}^\infty \frac{k^2-1}{k^3+4}$ Given the series 

$$\sum_{k=1}^\infty \frac{k^2-1}{k^3+4}.$$ 

I need to test for convergence/divergence. I can compare this to the series $\sum_{k=1}^\infty \frac{1}{k}$, which diverges. 
To use the comparison test, won't I need to show that $\frac{k^2-1}{k^3+4}>\frac{k^3}{k^4}=\frac{1}{k}$, in order to state the original series diverges?
This doesn't seem to hold, I feel like I'm missing the obvious. 
Any help is appreciated. 
Thanks. 

Hint: For $k\ge3$,
$$
\frac{k^2-1}{k^3+4}\ge\frac1{k+1}
$$
