Check convergence of $\sum_{n=1} \frac{n+1}{n^3+4n}$ I have to check convergence of $\sum_{n=1} \frac{n+1}{n^3+4n}$.
$$\sum_{n=1} \frac{n+1}{n^3+4n}\le \sum_{n=1} \frac{n+1}{n^3}=\sum_{n=1} \frac{1}{n^3}+\sum_{n=1} \frac{1}{n^2}$$
(both series on the right side are obviously convergent)

Still another way to check convergence:
$$\sum_{n\ge 1}\frac{n+1}{n^3+4n}\le \sum_{n\ge 1}\frac{2}{n^2+4}< 2\sum_{n\ge 1}\frac{1}{n^2}=\frac{\pi^2}{3}$$
