So I understand Euler's proof of the Basel Problem: $$\sum_{r=1}^{\infty}\frac{1}{r^2}= \frac{\pi^2}{6}$$
But how would I prove the general formula? Is it possible to prove without going into complex calculus and Fourier series?
$$\sum_{r=1}^{\infty}\frac{1}{r^{2k}}= \frac{(-1)^{k-1}\pi^{2k}2^{2k}B_{2k}}{2(2k)!}$$
Where $B_k = $ the $k^{th}$ Bernoulli number.