Closed form for $\sum_{k=1}^{\infty} \zeta(2k)-\zeta(2k+1)$

Question

From WolframAlpha it seems that $$ \frac{1}{2}=\sum_{k=1}^{\infty} \zeta(2k)-\zeta(2k+1) $$

Could someone provide a proof for this?

Thanks.

Answer 1

Writing zeta-functions as series and changing the summation order does the trick.