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: 
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:
$$\frac{k}{k^2-10}\ge\frac1k$$
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.