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In $1735$ Euler \cite{1} proved that for each positive integer $k$, theseries $\zeta(2k) = \sum_{\ell=1}^{\infty} \ell^{-2k}$ converges to a rationalmultiple of $\pi^{2k}$. Many demonstrations of this fact are now known, andEuler's discovery is traditionally proven using non-elementary techniques, suchas Fourier series or the calculus of residues \cite{2}. We give an elementaryproof, 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, yetarrived at a different formula, relating $\zeta(2k)$ to the Bernoulli numbers,\`a la Euler.

An elementary proof of the rationality of $\zeta(2n)/\pi^{2n}$