calculus - Prove that $sum frac{1}{n^2} = frac{pi^2}{6}$

In this answer two sequences are mentioned.
In particular, I would like to prove that

$$\sum_{n = 1}^{+ \infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

If I knew that the sequence converges to $\frac{\pi^2}{6}$, I could use the $\epsilon$-$M$ criterion to prove the convergence to that value.

But how to prove that the above sequence converges to that value if I don't know the value itself? Is there a general way to proceed in such cases?