Q:

Showing divergence or convergence, what is enough? ex $\sum_{30}^\infty \frac{n^2}{(n^3+5)^2}$

$$\sum_{30}^\infty \frac{n^2}{(n^3+5n)^2}$$
For example I have something like this can I just claim that only the leading terms matter and then show that I have $\frac{1}{n^4}$ or whatever and use the p series test or do I have to investigate the value of 5n? What is it was $5n^2?$ Would it matter at all since it only increases the amount of convergance? What if the numerator had more terms of a lesser degree? What if I had a subtraction of a lesser degree term on the bottom? 
The logic I see behind it is that only the highest degree terms matter as you approach infinity so it should be permissible to drop the lesser degree terms for a comparison to prove convergence or divergence. 

A:

Just note that,
$$ \frac{n^2}{(n^3+5n)^2} \sim \frac{n^2}{(n^3)^2} =\frac{1}{n^4}. $$
Now, you can make the comparison test.

