Question: Is the following series convergent or divergent? And how can I prove it? Which theorem should I use?

$$\sum\_{k=1}^{\infty}\dfrac{k}{k^2-10}$$

Answer: 
Hint: when $k$ is large, $\displaystyle\frac{k}{k^2-10} \geq \frac{k}{k^2}$.
The simple way is to use the asymptotic comparison with the harmonic series:

$$\frac{k}{k^2-10}\sim\_\infty\frac1k$$
or a simple comparison:
$$\frac{k}{k^2-10}\ge\frac1k$$