Does the series $\sum_{n=1}^\infty \frac{n+1}{n^3+10n}$ converge? Using the ratio test, we evaluate:
$$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty}\left| \frac{(n+1) + 1}{(n+1)^3 + 10(n+1)} \cdot \frac{n^3+10n}{n+1} \right| = \lim_{n \to \infty} \left| \frac{n^4 + 2n^3 + 10n^2 + 20n}{(n+1)^4 + 10(n+1)^2} \right|$$
$$ < \lim_{n \to \infty} \left| \frac{n^4 + 2n^3 + 10n^2 + 20n}{n^4 + 10n^2} \right| = 1$$
Hence $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$ and the series converges.
Is this an appropriate solution?

$$
\sum_{n=1}^\infty \frac{n+1}{n^3+10n} \leq \sum_{n=1}^\infty \frac{n+n}{n^3} =  2 \sum_{n=1}^\infty \frac{1}{n^2} < \infty.
$$
