In $1735$ Euler \cite{ 1} proved that for each positive integer $k$, the series $\zeta(2k) = \sum_{ \ell=1}^{ \infty} \ell^{ -2k}$ converges to a rational multiple of $\pi^{ 2k}$. Many demonstrations of this fact are now known, and Euler's discovery is traditionally proven using non-elementary techniques, such as Fourier series or the calculus of residues \cite{ 2}. We give an elementary proof, similar to Cauchy's \cite{ 3} proof of the identity $\zeta(2) = \pi^2/6$, only extended recursively for all values $\zeta(2k)$. Our main formula $$\zeta(2k)=-\dfrac{ (-\pi^2)^{ k}}{ 4^{ 2k}-4^{ k}}\left[\dfrac{ 4^{ k}k}{ (2k)!}+{ \displaystyle \sum_{ \ell=1}^{ k-1}}(4^{ 2\ell}-4^{ \ell})\dfrac{ 4^{ k-\ell}}{ (2k-2\ell)!}\dfrac{ \zeta(2\ell)}{ (-\pi^2)^{ \ell}}\right] \phantom{ spa}k = 1,2,3,\dots$$ may be derived from previously known formulae \cite{ 4}. Remarkably, Apostol \cite{ 5} discovered a proof similar to ours, yet arrived at a different formula, relating $\zeta(2k)$ to the Bernoulli numbers, \`a la Euler.
Categories: math.GM