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, $\displaystyle\frac{k}{k^2-10} \geq \frac{k}{k^2}$.
Integral Test 

$$ \lim\_{b\to\infty} \int\_{1}^b \frac{x}{x^2 - 1}\text{d}x$$

would lead to

$$\frac{1}{2} ln ({x^2-1}) $$

then just evaluate from b(infinity) to 1.

You will then see that the series diverges.