Prove the Series Converges or Diverges: $\sum_{n=1}^\infty \frac{5+2n}{(1+2n^2)^2}$ $$\sum_{n=1}^\infty \frac{5+2n}{(1+2n^2)^2}$$
How would I even do this to find if this converges or diverges? I tried doing the Telescoping series test but it just got really confusing. 
I got as far as this using partial fraction decomposition:
$$\sum_{n=1}^\infty \frac{5}{1+n^2} + \frac{2n-5n^2}{(1+2n^2)^2}$$
$$Sequence: (\frac{5}{2}-\frac{1}{3}),(1-\frac{16}{81}),(\frac{1}{2}
-\frac{29}{361}),...$$
I could really use some step by step instructions on how to do this one. Nothing seems to cancel out at all so I felt like I did it wrong. Did I even pick the right test to do it? This was on a quiz I took today so a thorough explanation wouldn't get me any points.

You may observe that, as $n \to \infty$,
$$
\frac{5+2n}{(1+2n^2)^2} \sim \frac{1}{2n^3}
$$ then use the comparison test.
