Another elementary proof of ${\sum }_{n\ge 1}1\mathrm{/}{n}^2={\pi }^2\mathrm{/}6$ and a recurrence formula for $\zeta (2k)$

In this shortnote, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values $:\zeta{(2 k +1)}$,$\zeta{(s)}$beingtheRiemannzetafunctionand$k$ a positive integer, is modified in a manner to furnish the even zeta values $\zeta (2k)$. As a result, I find an elementary proof of $\sum_{n=1}^\infty{{1/{n^2}}} = {\pi^2/6}$,aswellasarecurrenceformulafor$\zeta{(2k)}$ from which it follows that the ratio $\zeta (2k)\mathrm{/}{\pi }^{2k}$ is a rational number, without making use of Euler’s formula and Bernoulli numbers.